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Symbolic Dynamics and
Dynamical System Models

Topological, Categorical and Quantum
Dynamics

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Contents

Articles

I. C. Baianu, Ph.D., M.Inst.P., Editor (with listed contributors) i

Symbol 2

Systemics 4

Systems science 5

Systems theory 9

Systems analysis 20

Relational theory 22

Differential Equations 22

Computational physics 28

Dynamical Systems and Symbolic Dynamics 31

System 31

Dynamics 36

Dynamical systems theory 37

Symbolic dynamics 42

Molecular dynamics 44

Molecular modeling on GPU 56

Monte Carlo Methods 58

Quantum Dynamics 69

Mathematical Formulations of Quantum Dynamics 86

Quantum Chemistry and Biochemical Dynamics 95

List of quantum chemistry and solid state physics software 99

Basic Concepts in Symbolic Dynamics 103

Sequential dynamical system 103

Automata theory 104

Time series analysis 108

Lag operator 112

Shift operator 114

Shift space 115

Markov partition 116

Sharkovskii's theorem 118

Ergodic system 119

Ergodic theory 125

Measure-preserving dynamical system 130

Periodic orbit 133

Hilbert space 136

Spherical harmonics 162

Quantum computer 177

Topological Quantum Computers 186

Categorical and Topological Dynamics. Category Theory and Categorical

Dynamics Concepts 189

Algebraic Geometry 189

Category theory 197

Higher-dimensional algebra 204

Higher category theory 208

Algebraic topology 210

Topological dynamics 214

Graph dynamical system 215

Analysis of Systems 218

Dynamic Bayesian network 220

Dynamic network analysis 220

Dynamic circuit network 222

Tensor product network 223

Cybernetics 224

Systems Biology 232

Neurosciences 239

Biocybernetics 247

Computational neuroscience 248

Molecular Neurosciences and Molecular Medicine 256

Complex Systems 257

Complex Systems Biology 261

Mathematical, Relational and Theoretical Biology 266

Lotka— Volterra Differential Equations in Population Biology 275

Chaotic Dynamics 282

Chaos theory 282

Chaos theory in organizational development 295

Attractor 298

Lorenz attractor 303

Strange attractor 307

Butterfly effect 312

Standard map 315

Henon map 319

Horseshoe map 321

Coupled map lattice 326

List of chaotic maps 33 1

Chua's circuit 333

Double pendulum 334

Dynamical billiards 338

Bifurcation theory 344

Rossler attractor 347

Synchronizing Chaos 355

The Possibility of Quantum Chaos ? 357

Fractals and Fractional Dimensions 366

Mandelbrot set 373

Julia set 391

Complexity 401

At the Edge of Chaos 407

Chaos control 408

Butterfly effect 409

Applications 413

Datastorage 413

Data transmission 413

Related Biographies 418

Isaac Newton 418

Bernhard Riemann 438

Jean Dieudonne 442

Alexander Grothendieck 445

Charles Ehresmann 453

Samuel Eilenberg 455

Emil Artin 457

Ronald Brown 467

Henri Poincare 470

Henri Cartan 485

Jacques Hadamard 488

Niels Bohr 491

Werner Heisenberg 500

Albert Einstein 522

Emmy Noether 547

Norbert Wiener 572

Paul Dirac 578

John von Neumann 588

George Birkhoff 602

Stephen Weinberg 605

Claude Shannon 610

Ludwig von Bertalanffy 619

Stephen Smale 624

Yakov Sinai 627

Marston Morse 629

G. A. Hedlund 630

William Ross Ashby 63 1

Robert Rosen 634

Edward Norton Lorenz 635

Otto Rossler 638

Paul Koebe 640

Jakob Nielsen (mathematician) 641

Benoit Mandelbrot 643

References

Article Sources and Contributors 649

Image Sources, Licenses and Contributors 664

Article Licenses

License 671

I. C. Baianu, Ph.D., M.Inst.P., Editor (with

listed contributors)

Symbol

A symbol is something such as an particular mark that represents some piece of information. For example, a red
octagon may be a symbol for "STOP". On maps, tables would mean campsite. Numerals are symbols for numbers
(amounts). All language consists of symbols. Personal names are symbols representing individuals.

Psychoanalysis and archetypes

Swiss psychoanalyst Carl Jung, who studied archetypes, proposed an alternative definition of symbol, distinguishing
it from the term sign. In Jung's view, a sign stands for something known, as a word stands for its referent. He
contrasted this with symbol, which he used to stand for something that is unknown and that cannot be made clear or
precise. An example of a symbol in this sense is Christ as a symbol of the archetype called self. For example,
written languages are composed of a variety of different symbols that create words. Through these written words,
humans communicate with each other. Kenneth Burke described Homo sapiens as a "symbol-using, symbol making,
and symbol misusing animal" to indicate that a person creates symbols in her or his life as well as misuses them. One
example he uses to indicate his meaning behind symbol misuse is the story of a man who, when told a particular
food item was whale blubber, could barely keep from throwing it up. Later, his friend discovered it was actually just
a dumpling. But the man's reaction was a direct consequence of the symbol of "blubber" representing something
inedible in his mind. In addition, the symbol of "blubber" for the man was created by him through various kinds of
learning. Burke emphasizes that humans gain this type of learning that helps us create symbols by seeing various
print sources, our life experiences, and symbols about the past.

Burke goes on to describe symbols as also being derived from Sigmund Freud's work on condensation and
displacement further stating that they are not just relevant to the theory of dreams, but also to "normal symbol
systems". He says they are related through "substitution" where one word, phrase, or symbol is substituted for
another in order to change the meaning. In other words, if a person does not understand a certain word or phrase,
another person may substitute a synonym or symbol in order to get the meaning of the original word or phrase
across. However, when faced with that new way of interpreting a specific symbol, a person may change their already
formed ideas to incorporate the new information based on how the symbol is expressed to the person.

Jean Dalby Clift says that people not only add their own interpretations to symbols, they also create personal
symbols that represent their own understanding of their lives: what she calls "core images" of the person. She argues
that symbolic work with these personal symbols or core images can be as useful as working with dream symbols in
psychoanalysis or counseling.

Etymology

The word symbol came to the English language by way of Middle English, from Old French, from Latin, from the
Greek oii^poXov (symbolon) from the root words aw- (syn-), meaning "together," and poXr] (bole), "a throw",
having the approximate meaning of "to throw together", literally a "co-incidence", also "sign, ticket, or contract".
The earliest attestation of the term is in the Homeric Hymn to Hermes where Hermes on seeing the tortoise exclaims
ovjijioXov ijSrj /xoi fiey' ovrjOLjiov "symbolon [symbol/sign/portent/encounter/chance find?] of joy to me!" before
turning it into a lyre.

Symbol

Role of context in symbolism

A symbol's meaning may be modified by various factors including popular usage, history, and contextual intent.

Historical meaning

This history of a symbol is one of many factors in determining a particular symbol's apparent meaning.
Consequently, symbols with emotive power carry problems analogous to false etymologies.

For example, the Rebel Flag of the American South predates the American Civil War. An early variant of the crossed
bars resembled the Scottish Flag.

Juxtaposition

Juxtaposition further complicates the matter. Similar five— pointed stars might signify a law enforcement officer or
a member of the armed services, depending the uniform.

Notes

[1] Psychological Types, C. G. Jung, (trans. Baynes), p. 601.

[2] Jean Dalby Clift, Core Images of the Self: A Symbolic Approach to Healing and Wholeness. Crossroad, 1992.

[3] http://books. google. com/books?hl=en&lr=&id=ERsyiUOYI4kC&oi=fnd&pg=PA15&dq=confederate+flag+extremist+groups+ku+

klux+klan&ots=7IgVGRosTS&sig=YhygSgtlzsU_fFvgxzTHNHZPUI
[4] http://en.wiktionary.org/wiki/juxtaposition

External links

symbols.com (http://www.symbols.com/)

Symbolism Wiki

Symbols & signs (http://www.digiden.nl/en/symbols-and-signs/)

Ancient Symbolism (http://www.ancient-symbols.com/)

Numericana (http://www.numericana.com/answer/symbol.htm)

logo search (http://www.logo-search.com)

Systemics

In the context of systems science and systems philosophy, the term systemics refers to an initiative to study systems
from a holistic point of view. It is an attempt at developing logical, mathematical, engineering and philosophical
paradigms and frameworks in which physical, technological, biological, social, cognitive, and metaphysical systems
can be studied and modeled.

The term "systemics" was coined in the 1970s by Mario Bunge and others, as an alternative paradigm for research
related to general systems theory and systems science.

References

[1] Mario Bunge (1979). A world of systems. Dordrecht; Boston, Reidel.

Further reading

• Mario Bunge (1979), A world of systems. Dordrecht; Boston, Reidel.

• Charles Francois (1999), Systemics and Cybernetics in a Historical Perspective (http://www.uni-klu.ac.at/
~gossimit/ifsr/francois/papers/systemics_and_cybernetics_in_a_historical_perspective.pdf). in: Systems
Research and Behavioral Science, Vol 16, pp. 203—219.

• Watson, D. E., G. E. Schwartz, L. G. S. Russek (1998), The Theory of Enformed Systems - A Paradigm of
Organization and Holistic Systems (http://www.enformy.com/$wsr02.html)

• Donald E. Watson (2005), Systemics: The Most Basic Science (http://www.enformy.com/$system.html).

• Frederic Vester (2008), The Art of interconnected thinking: Tools and concepts for a new approach to tackling
complexity; Munich, MCB.

External links

• A Taste of Systemics (http://www.isss.org/taste.html) By Bela H. Banathy

• Journal of Systemics, Cybernetics and Informatics (http://www.iiisci.org/Journal/SCI/Home.asp)

• Computational Philosophy of Science (http://mitpress.mit. edu/catalog/item/default.asp?ttype=2&tid=5526) -
The MIT Press

Systems science

Systems science is an interdisciplinary field of science
that studies the nature of complex systems in nature,
society, and science. It aims to develop interdisciplinary
foundations, which are applicable in a variety of areas,
such as engineering, biology, medicine and social
sciences.

Systems sciences covers formal sciences fields like
complex systems, cybernetics, dynamical systems theory,
and systems theory, and applications in the field of the
natural and social sciences and engineering, such as
control theory, operations research, social systems theory,
systems biology, systems dynamics, systems ecology,
systems engineering and systems psychology.

Theories

Impression of systems thinking about society

[1]

Since the emergence of the General Systems Research in

the 1950s systems thinking and systems science has been developed into all kinds of theoretical frameworks. The

following overview will only show the most basic types.

Systems analysis

Systems analysis is the interdisciplinary branch of
science, dealing with analysis of systems, often
prior to their automation as computer systems, and
the interactions within those systems. This field is
closely related to operations research.

Systems design

In computing systems design is the process or art of
defining the hardware and software architecture,
components, modules, interfaces, and data for a
computer system to satisfy specified requirements.
One could see it as the application of systems
theory to computing. Some overlap with the
discipline of systems analysis appears inevitable.

System dynamics

System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with
internal feedback loops and time delays that affect the behaviour of the entire system. What makes using
system dynamics different from other approaches to studying complex systems is the use of feedback loops
and stocks and flows. These elements help describe how even seemingly simple systems display baffling
nonlinearity.

Systems engineering

Systems Engineering (SE) is an interdisciplinary field of engineering, that focuses on the development and
organization of complex artificial systems. Systems engineering has emerged into all kinds of sciences, and

Systems science

universities nowadays offer all kinds of specialized academic programs.
Systems Methodologies

There are several types of Systems Methodologies, that is, disciplines for analysis of systems. For example:

• Soft Systems Methodology (SSM) : in the field of organizational studies is an approach to organisational
process modelling and it can be used both for general problem solving and in the management of change. It
was developed in England by academics at the University of Lancaster Systems Department through a ten year
Action Research programme.

• System Development Methodology (SDM) in the field of IT development is a general term applied to a variety
of structured, organized processes for developing information technology and embedded software systems.

Systems theories

Systems theory is an interdisciplinary field of science. It studies the nature of complex systems in nature,
society, and science. More specificially, it is a framework by which one can analyze and/or describe any group
of objects that work in concert to produce some result.

Systems science

Systems sciences are scientific disciplines partly based on systems thinking such as Chaos theory, Complex
systems, Control theory, Cybernetics, Sociotechnical systems theory, Systems biology, Systems ecology,
Systems psychology and the already mentioned Systems dynamics, Systems engineering and Systems theory.

Fields

Systems sciences covers formal sciences fields like dynamical systems theory and applications in the field of the
natural and social sciences and engineering, such as social systems theory and systems dynamics.

Chaos theory

Complex systems

Complexity theory

Cybernetics

Biocybernetics
Engineering cybernetics
Management cybernetics
Medical cybernetics
New Cybernetics
Second-order cybernetics

Operations research

Systems biology

• Computational systems biology

• Synthetic biology

• Systems immunology

System dynamics

• Social dynamics

Systems ecology

• Ecosystem ecology

Systems psychology

• Ergonomics

• Family systems theory

• Systemic therapy
Systems theory

Biochemical systems theory
Ecological systems theory
Developmental systems theory
General systems theory
Living systems theory
LTI system theory
Sociotechnical systems theory
Mathematical system theory
World-systems theory
Systems theory in sociology

• Talcott Parsons

• Niklas Luhmann

Systems science

Control theory • Systems engineering

• Affect control theory • Biological systems engineering

• Control engineering • Earth systems engineering and management

• Control systems • Enterprise systems engineering

• Dynamical systems • Systems analysis

Perceptual control theory

Systems theory in anthropology

Systems scientists

General systems scientists can be divided into three generations. The founders of the systems movement like Ludwig
von Bertalanffy, Kenneth Boulding, Ralph Gerard, James Grier Miller, George J. Klir,and Anatol Rapoport were all
born between 1900 and 1920. They all came from different natural and social science disciplines and joined forces in
the 1950s to established the general systems theory paradigm. Along with the organization of their efforts a first
generation of systems scientists rose.

Among them were other scientists like Ackoff, Ashby and Churchman, who popularized the systems concept in the
1950s and 1960s. These scientists inspired and educated a second generation with more notable scientist like Ervin
Laszlo (1932) and Fritjof Capra (1939), who wrote about systems theory in the 1970s and 1980s. Others got
acquainted and started studying these works in the 1980s and started writing about it since the 1990s. Debora
Hammond can be seen as a typical representative of these third generation of general systems scientists.

Organizations

The International Society for the Systems Sciences (ISSS) is an organisation for interdisciplinary collaboration and
synthesis of systems sciences. The ISSS is unique among systems-oriented institutions in terms of the breadth of its
scope, bringing together scholars and practitioners from academic, business, government, and non-profit
organizations. Based on fifty years of tremendous interdisciplinary research from the scientific study of complex
systems to interactive approaches in management and community development. This society was initially conceived
in 1954 at the Stanford Center for Advanced Study in the Behavioral Sciences by Ludwig von Bertalanffy, Kenneth
Boulding, Ralph Gerard, and Anatol Rapoport.

In the field of systems science the International Federation for Systems Research (IFSR) is an international
federation for global and local societies in the field of systems science. This federation is a non-profit, scientific and
educational agency founded in 1981, and constituted of some thirty member organizations from various countries.
The overall purpose of this Federation is to advance cybernetic and systems research and systems applications and to
serve the international systems community.

The best known research institute in the field is the Santa Fe Institute (SFI) located in Santa Fe, New Mexico, United
States, dedicated to the study of complex systems. This institute was founded in 1984 by George Cowan, David
Pines, Stirling Colgate, Murray Gell-Mann, Nick Metropolis, Herb Anderson, Peter A. Carruthers, and Richard
Slansky. All but Pines and Gell-Mann were scientists with Los Alamos National Laboratory. SFI's original mission
was to disseminate the notion of a separate interdisciplinary research area, complexity theory referred to at SFI as
complexity science.

Systems science

References

[1] Illustration is made by Marcel Douwe Dekker (2007) based on an own standard and Pierre Malotaux (1985), "Constructieleer van de

mensenlijke samenwerking", in BB5 Collegedictaat TU Delft, pp. 120-147.
[2] MIT System Dynamics in Education Project (SDEP) (http://sysdyn.clexchange.org)
[3] See for further details: List of systems engineering at universities

Further reading

• B. A. Bayraktar, Education in Systems Science, 1979, 369 pp.

• Kenneth D. Bailey, "Fifty Years of Systems Science: Further Reflections", Systems Research and Behavioral
Science, 22, 2005, pp. 355-361.

• Robert L. Flood, Ewart R Carson, Dealing with Complexity: An Introduction to the Theory and Application of
Systems Science, 1988.

• George J. Klir, Facets of Systems Science, Plenum Press, 1991.

• Ervin Laszlo, Systems Science and World Order: Selected Studies, 1983.

• Anatol Rapoport (ed.), General Systems: Yearbook of the Society for the Advancement of General Systems
Theory, Society for General Systems Research, Vol 1., 1956.

• Li D. Xu, "The contributions of Systems Science to Information Systems Research", Systems Research and
Behavioral Science, 17, 2000, pp. 105—116.

• Graeme Donald Snooks, "A general theory of complex living systems: Exploring the demand side of dynamics".
Complexity, vol. 13, no. 6, July/August 2008.

• John N. Warfield, "A proposal for Systems Science", Systems Research and Behavioral Science, 20, 2003,
pp. 507-520.

External links

• Principia Cybernetica Web (http://pespmcl.vub.ac.be/)

• International Society for the System Sciences (http://isss.org/world/)

• UK Systems Society (http://www.ukss.org.uk)

Systems theory

Systems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that can
be applied to all types of systems in all fields of research. The term does not yet have a well-established, precise
meaning, but systems theory can reasonably be considered a specialization of systems thinking and a generalization
of systems science. The term originates from Bertalanffy's General System Theory (GST) and is used in later efforts
in other fields, such as the action theory of Talcott Parsons and the system-theory of Niklas Luhmann.

In this context the word "systems" is used to refer specifically to self-regulating systems, i.e. that are self-correcting
through feedback. Self-regulating systems are found in nature, including the physiological systems of our body, in
local and global ecosystems, and in climate.

Overview

Contemporary ideas from systems theory have grown with

diversified areas, exemplified by the work of Bela H. Banathy,

ecological systems with Howard T. Odum, Eugene Odum and Fritjof

Capra, organizational theory and management with individuals such

as Peter Senge, interdisciplinary study with areas like Human

Resource Development from the work of Richard A. Swanson, and

insights from educators such as Debora Hammond and Alfonso

Montuori. As a transdisciplinary, interdisciplinary and

multiperspectival domain, the area brings together principles and

concepts from ontology, philosophy of science, physics, computer

science, biology, and engineering as well as geography, sociology,

political science, psychotherapy (within family systems therapy) and

economics among others. Systems theory thus serves as a bridge for interdisciplinary dialogue between autonomous

areas of study as well as within the area of systems science itself.

In this respect, with the possibility of misinterpretations, von Bertalanffy believed a general theory of systems
"should be an important regulative device in science," to guard against superficial analogies that "are useless in
science and harmful in their practical consequences." Others remain closer to the direct systems concepts developed
by the original theorists. For example, Ilya Prigogine, of the Center for Complex Quantum Systems at the University
of Texas, Austin, has studied emergent properties, suggesting that they offer analogues for living systems. The
theories of autopoiesis of Francisco Varela and Humberto Maturana are a further development in this field.
Important names in contemporary systems science include Russell Ackoff, Bela H. Banathy, Anthony Stafford Beer,
Peter Checkland, Robert L. Flood, Fritjof Capra, Michael C. Jackson, Edgar Morin and Werner Ulrich, among
others.

Margaret Mead was an influential figure in systems
theory.

With the modern foundations for a general theory of systems following the World Wars, Ervin Laszlo, in the preface

for Bertalanffy's book Perspectives on General System Theory, maintains that the translation of "general system

121
theory" from German into English has "wrought a certain amount of havoc". The preface explains that the original

concept of a general system theory was "Allgemeine Systemtheorie (or Lehre)", pointing out the fact that "Theorie"

(or "Lehre") just as "Wissenschaft" (translated Scholarship), "has a much broader meaning in German than the

i 121

closest English words theory and science'". With these ideas referring to an organized body of knowledge and

"any systematically presented set of concepts, whether they are empirical, axiomatic, or philosophical, "Lehre" is

associated with theory and science in the etymology of general systems, but also does not translate from the German

12]

very well; "teaching" is the "closest equivalent", but "sounds dogmatic and off the mark". While many of the root
meanings for the idea of a "general systems theory" might have been lost in the translation and many were led to

Systems theory 10

believe that the systems theorists had articulated nothing but a pseudoscience, systems theory became a
nomenclature that early investigators used to describe the interdependence of relationships in organization by
defining a new way of thinking about science and scientific paradigms.

A system from this frame of reference is composed of regularly interacting or interrelating groups of activities. For
example, in noting the influence in organizational psychology as the field evolved from "an individually oriented
industrial psychology to a systems and developmentally oriented organizational psychology," it was recognized that
organizations are complex social systems; reducing the parts from the whole reduces the overall effectiveness of
organizations. This is different from conventional models that center on individuals, structures, departments and
units separate in part from the whole instead of recognizing the interdependence between groups of individuals,
structures and processes that enable an organization to function. Laszlo explains that the new systems view of
organized complexity went "one step beyond the Newtonian view of organized simplicity" in reducing the parts from
the whole, or in understanding the whole without relation to the parts. The relationship between organizations and
their environments became recognized as the foremost source of complexity and interdependence. In most cases the
whole has properties that cannot be known from analysis of the constituent elements in isolation. Bela H. Banathy,
who argued — along with the founders of the systems society — that "the benefit of humankind" is the purpose of
science, has made significant and far-reaching contributions to the area of systems theory. For the Primer Group at
ISSS, Banathy defines a perspective that iterates this view:

The systems view is a world-view that is based on the discipline of SYSTEM INQUIRY. Central to systems
inquiry is the concept of SYSTEM. In the most general sense, system means a configuration of parts
connected and joined together by a web of relationships. The Primer group defines system as a family of
relationships among the members acting as a whole. Von Bertalanffy defined system as "elements in standing
relationship.
_[5]

Similar ideas are found in learning theories that developed from the same fundamental concepts, emphasizing how
understanding results from knowing concepts both in part and as a whole. In fact, Bertalanffy's organismic
psychology paralleled the learning theory of Jean Piaget. Interdisciplinary perspectives are critical in breaking
away from industrial age models and thinking where history is history and math is math, the arts and sciences
specialized and separate, and where teaching is treated as behaviorist conditioning. The influential contemporary

ro]

work of Peter Senge provides detailed discussion of the commonplace critique of educational systems grounded in
conventional assumptions about learning, including the problems with fragmented knowledge and lack of holistic
learning from the "machine-age thinking" that became a "model of school separated from daily life." It is in this way
that systems theorists attempted to provide alternatives and an evolved ideation from orthodox theories with
individuals such as Max Weber, Emile Durkheim in sociology and Frederick Winslow Taylor in scientific
management, which were grounded in classical assumptions. The theorists sought holistic methods by developing
systems concepts that could be integrated with different areas.

The contradiction of reductionism in conventional theory (which has as its subject a single part) is simply an
example of changing assumptions. The emphasis with systems theory shifts from parts to the organization of parts,
recognizing interactions of the parts are not "static" and constant but "dynamic" processes. Conventional closed
systems were questioned with the development of open systems perspectives. The shift was from absolute and
universal authoritative principles and knowledge to relative and general conceptual and perceptual knowledge,
still in the tradition of theorists that sought to provide means in organizing human life. Meaning, the history of ideas
that preceded were rethought not lost. Mechanistic thinking was particularly critiqued, especially the industrial-age
mechanistic metaphor of the mind from interpretations of Newtonian mechanics by Enlightenment philosophers and
later psychologists that laid the foundations of modern organizational theory and management by the late 19th
century. Classical science had not been overthrown, but questions arose over core assumptions that historically
influenced organized systems, within both social and technical sciences.

Systems theory 1 1

History

Timeline

Precursors

• Saint-Simon (1760-1825), Karl Marx (1817-1883), Herbert Spencer (1820-1903), Rudolf Clausius (1822-1888), Vilfredo Pareto
(1848-1923), Emile Durkheim (1858-1917), Alexander Bogdanov (1873-1928), Nicolai Hartmann (1882-1950), Stafford Beer (1926-2002),
Robert Maynard Hutchins (1929—1951), among others

Pioneers

• 1946-1953 Macy conferences

• 1948 Norbert Wiener publishes Cybernetics or Control and Communication in the Animal and the Machine

• 1954 Ludwig von Bertalanffy, Anatol Rapoport, Ralph W. Gerard, Kenneth Boulding establish Society for the Advancement of General
Systems Theory, in 1956 renamed to Society for General Systems Research.

• 1955 W. Ross Ashby publishes Introduction to Cybernetics

• 1968 Ludwig von Bertalanffy publishes General System theory: Foundations, Development, Applications

Developments

1970-1980s Second-order cybernetics developed by Heinz von Foerster, Gregory Bateson, Humberto Maturana and others

1971-1973 Cybersyn, rudimentary internet and cybernetic system for democratic economic planning developed in Chile under Allende

government by Stafford Beer

1970s Catastrophe theory (Rene Thom, E.C. Zeeman) Dynamical systems in mathematics.

1977 Ilya Prigogine received the Nobel Prize for his works on self-organization, conciliating important systems theory concepts with system

thermodynamics.

1980s Chaos theory David Ruelle, Edward Lorenz, Mitchell Feigenbaum, Steve Smale, James A. Yorke

1986 Context theory, Anthony Wilden

1988 International Society for Systems Science

1990 Complex adaptive systems (CAS), John H. Holland, Murray Gell-Mann, W. Brian Arthur

Contemporary applications

[12]

• 2010 Elements of the public policy of science, technology and innovation , Julio E. Rubio

Whether considering the first systems of written communication with Sumerian cuneiform to Mayan numerals, or
the feats of engineering with the Egyptian pyramids, systems thinking in essence dates back to antiquity.

Differentiated from Western rationalist traditions of philosophy, C. West Churchman often identified with the I

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Ching as a systems approach sharing a frame of reference similar to pre-Socratic philosophy and Heraclitus. Von

Bertalanffy traced systems concepts to the philosophy of G.W. von Leibniz and Nicholas of Cusa's coincidentia

oppositorum. While modern systems are considerably more complicated, today's systems are embedded in history.

An important step to introduce the systems approach, into (rationalist) hard sciences of the 19th century, was the
energy transformation, by figures like James Joule and Sadi Carnot. Then, the Thermodynamic of this century, with
Rudolf Clausius, Josiah Gibbs and others, built the system reference model, as a formal scientific object.

Systems theory as an area of study specifically developed following the World Wars from the work of Ludwig von
Bertalanffy, Anatol Rapoport, Kenneth E. Boulding, William Ross Ashby, Margaret Mead, Gregory Bateson, C.
West Churchman and others in the 1950s, specifically catalyzed by the cooperation in the Society for General
Systems Research. Cognizant of advances in science that questioned classical assumptions in the organizational
sciences, Bertalanffy's idea to develop a theory of systems began as early as the interwar period, publishing 'An
Outline for General Systems Theory" in the British Journal for the Philosophy of Science, Vol 1, No. 2, by 1950.
Where assumptions in Western science from Greek thought with Plato and Aristotle to Newton's Principia have
historically influenced all areas from the hard to social sciences (see David Easton's seminal development of the
"political system" as an analytical construct), the original theorists explored the implications of twentieth century
advances in terms of systems.

Subjects like complexity, self-organization, connectionism and adaptive systems had already been studied in the
1940s and 1950s. In fields like cybernetics, researchers like Norbert Wiener, William Ross Ashby, John von

Systems theory 12

Neumann and Heinz von Foerster examined complex systems using mathematics. John von Neumann discovered
cellular automata and self-reproducing systems, again with only pencil and paper. Aleksandr Lyapunov and Jules
Henri Poincare worked on the foundations of chaos theory without any computer at all. At the same time Howard T.
Odum, the radiation ecologist, recognised that the study of general systems required a language that could depict
energetics, thermodynamic and kinetics at any system scale. Odum developed a general systems, or Universal
language, based on the circuit language of electronics to fulfill this role, known as the Energy Systems Language.
Between 1929-1951, Robert Maynard Hutchins at the University of Chicago had undertaken efforts to encourage
innovation and interdisciplinary research in the social sciences, aided by the Ford Foundation with the

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interdisciplinary Division of the Social Sciences established in 1931. Numerous scholars had been actively
engaged in ideas before (Tectology of Alexander Bogdanov published in 1912-1917 is a remarkable example), but in
1937 von Bertalanffy presented the general theory of systems for a conference at the University of Chicago.

The systems view was based on several fundamental ideas. First, all phenomena can be viewed as a web of
relationships among elements, or a system. Second, all systems, whether electrical, biological, or social, have
common patterns, behaviors, and properties that can be understood and used to develop greater insight into the
behavior of complex phenomena and to move closer toward a unity of science. System philosophy, methodology and
application are complementary to this science. By 1956, the Society for General Systems Research was
established, renamed the International Society for Systems Science in 1988. The Cold War affected the research
project for systems theory in ways that sorely disappointed many of the seminal theorists. Some began to recognize
theories defined in association with systems theory had deviated from the initial General Systems Theory (GST)
view. The economist Kenneth Boulding, an early researcher in systems theory, had concerns over the

manipulation of systems concepts. Boulding concluded from the effects of the Cold War that abuses of power always
prove consequential and that systems theory might address such issues. Since the end of the Cold War, there has
been a renewed interest in systems theory with efforts to strengthen an ethical view.

Developments in system theories
General systems research and systems inquiry

Many early systems theorists aimed at finding a general systems theory that could explain all systems in all fields of
science. The term goes back to Bertalanffy's book titled "General System theory: Foundations, Development,
Applications" from 1968. According to Von Bertalanffy, he developed the "allgemeine Systemlehre" (general

ri7i

systems teachings) first via lectures beginning in 1937 and then via publications beginning in 1946.

Von Bertalanffy's objective was to bring together under one heading the organismic science that he had observed in
his work as a biologist. His desire was to use the word "system" to describe those principles which are common to
systems in general. In GST, he writes:

...there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of
their particular kind, the nature of their component elements, and the relationships or "forces" between them. It
seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles
applying to systems in general.

Ervin Laszlo in the preface of von Bertalanffy's book Perspectives on General System Theory.

Thus when von Bertalanffy spoke of Allgemeine Systemtheorie it was consistent with his view that he was
proposing a new perspective, a new way of doing science. It was not directly consistent with an interpretation
often put on "general system theory", to wit, that it is a (scientific) "theory of general systems." To criticize it
as such is to shoot at straw men. Von Bertalanffy opened up something much broader and of much greater
significance than a single theory (which, as we now know, can always be falsified and has usually an
ephemeral existence): he created a new paradigm for the development of theories.

Systems theory 13

Ludwig von Bertalanffy outlines systems inquiry into three major domains: Philosophy, Science, and Technology. In
his work with the Primer Group, Bela H. Banathy generalized the domains into four integratable domains of
systemic inquiry:

Domain Description

Philosophy the ontology, epistemology, and axiology of systems;

Theory a set of interrelated concepts and principles applying to all systems

Methodology the set of models, strategies, methods, and tools that instrumentalize systems theory and philosophy

Application the application and interaction of the domains

These operate in a recursive relationship, he explained. Integrating Philosophy and Theory as Knowledge, and

1211

Method and Application as action, Systems Inquiry then is knowledgeable action.

Cybernetics

The term cybernetics derives from a Greek word which meant steersman, and which is the origin of English words
such as "govern". Cybernetics is the study of feedback and derived concepts such as communication and control in
living organisms, machines and organisations. Its focus is how anything (digital, mechanical or biological) processes
information, reacts to information, and changes or can be changed to better accomplish the first two tasks.

The terms "systems theory" and "cybernetics" have been widely used as synonyms. Some authors use the term
cybernetic systems to denote a proper subset of the class of general systems, namely those systems that include
feedback loops. However Gordon Pask's differences of eternal interacting actor loops (that produce finite products)
makes general systems a proper subset of cybernetics. According to Jackson (2000), von Bertalanffy promoted an
embryonic form of general system theory (GST) as early as the 1920s and 1930s but it was not until the early 1950s
it became more widely known in scientific circles.

Threads of cybernetics began in the late 1800s that led toward the publishing of seminal works (e.g., Wiener's
Cybernetics in 1948 and von Bertalanffy's General Systems Theory in 1968). Cybernetics arose more from
engineering fields and GST from biology. If anything it appears that although the two probably mutually influenced
each other, cybernetics had the greater influence. Von Bertalanffy (1969) specifically makes the point of
distinguishing between the areas in noting the influence of cybernetics: "Systems theory is frequently identified with
cybernetics and control theory. This again is incorrect. Cybernetics as the theory of control mechanisms in
technology and nature is founded on the concepts of information and feedback, but as part of a general theory of
systems;" then reiterates: "the model is of wide application but should not be identified with 'systems theory' in
general", and that "warning is necessary against its incautious expansion to fields for which its concepts are not
made." (17-23). Jackson (2000) also claims von Bertalanffy was informed by Alexander Bogdanov's three volume
Tectology that was published in Russia between 1912 and 1917, and was translated into German in 1928. He also
states it is clear to Gorelik (1975) that the "conceptual part" of general system theory (GST) had first been put in
place by Bogdanov. The similar position is held by Mattessich (1978) and Capra (1996). Ludwig von Bertalanffy
never even mentioned Bogdanov in his works, which Capra (1996) finds "surprising".

Cybernetics, catastrophe theory, chaos theory and complexity theory have the common goal to explain complex
systems that consist of a large number of mutually interacting and interrelated parts in terms of those interactions.
Cellular automata (CA), neural networks (NN), artificial intelligence (AI), and artificial life (ALife) are related
fields, but they do not try to describe general (universal) complex (singular) systems. The best context to compare
the different "C"-Theories about complex systems is historical, which emphasizes different tools and methodologies,
from pure mathematics in the beginning to pure computer science now. Since the beginning of chaos theory when
Edward Lorenz accidentally discovered a strange attractor with his computer, computers have become an
indispensable source of information. One could not imagine the study of complex systems without the use of

Systems theory 14

computers today.

Complex adaptive systems

Complex adaptive systems are special cases of complex systems. They are complex in that they are diverse and made
up of multiple interconnected elements and adaptive in that they have the capacity to change and learn from
experience. The term complex adaptive systems was coined at the interdisciplinary Santa Fe Institute (SFI), by John
H. Holland, Murray Gell-Mann and others. However, the approach of the complex adaptive systems does not take
into account the adoption of information which enables people to use it.

CAS ideas and models are essentially evolutionary. Accordingly, the theory of complex adaptive systems bridges
developments of the system theory with the ideas of 'generalized Darwinism', which suggests that Darwinian
principles of evolution help explain a wide range of phenomena.

Applications of system theories
Living systems theory

Living systems theory is an offshoot of von Bertalanffy's general systems theory, created by James Grier Miller,
which was intended to formalize the concept of "life". According to Miller's original conception as spelled out in his
magnum opus Living Systems, a "living system" must contain each of 20 "critical subsystems", which are defined by
their functions and visible in numerous systems, from simple cells to organisms, countries, and societies. In Living
Systems Miller provides a detailed look at a number of systems in order of increasing size, and identifies his
subsystems in each.

James Grier Miller (1978) wrote a 1,102 pages volume to present his living systems theory. He constructed a general
theory of living systems by focusing on concrete systems — nonrandom accumulations of matter-energy in physical
space-time organized into interacting, interrelated subsystems or components. Slightly revising the original model a
dozen years later, he distinguished eight "nested" hierarchical levels in such complex structures. Each level is
"nested" in the sense that each higher level contains the next lower level in a nested fashion.

Organizational theory

Systems theory

The systems framework is also fundamental to organizational theory as
organizations are complex dynamic goal-oriented processes. One of the early
thinkers in the field was Alexander Bogdanov, who developed his Tectology, a
theory widely considered a precursor of von Bertalanffy's GST, aiming to model
and design human organizations (see Mattessich 1978, Capra 1996). Kurt Lewin
was particularly influential in developing the systems perspective within
organizational theory and coined the term "systems of ideology", from his

frustration with behavioral psychologies that became an obstacle to sustainable

T221
work in psychology. Jay Forrester with his work in dynamics and management

alongside numerous theorists including Edgar Schein that followed in their

tradition since the Civil Rights Era have also been influential.

The systems to organizations relies heavily upon achieving negative entropy

through openness and feedback. A systemic view on organizations is

transdisciplinary and integrative. In other words, it transcends the perspectives of

individual disciplines, integrating them on the basis of a common "code", or more

exactly, on the basis of the formal apparatus provided by systems theory. The

systems approach gives primacy to the interrelationships, not to the elements of the

system. It is from these dynamic interrelationships that new properties of the

system emerge. In recent years, systems thinking has been developed to provide techniques for studying systems in

holistic ways to supplement traditional reductionistic methods. In this more recent tradition, systems theory in

organizational studies is considered by some as a humanistic extension of the natural sciences.

Kurt Lewin attended the Macy

conferences and is commonly

identified as the founder of the

movement to study groups

scientifically.

Software and computing

In the 1960s, systems theory was adopted by the post John Von Neumann computing and information technology
field and, in fact, formed the basis of structured analysis and structured design (see also Larry Constantine, Tom
DeMarco and Ed Yourdon). It was also the basis for early software engineering and computer-aided software
engineering principles.

By the 1970s, General Systems Theory (GST) was the fundamental underpinning of most commercial software
design techniques, and by the 1980, W. Vaughn Frick and Albert F. Case, Jr. had used GST to design the "missing
link" transformation from system analysis (defining what's needed in a system) to system design (what's actually
implemented) using the Yourdon/DeMarco notation. These principles were incorporated into computer-aided
software engineering tools delivered by Nastec Corporation, Transform Logic, Inc., KnowledgeWare (see Fran
Tarkenton and James Martin), Texas Instruments, Arthur Andersen and ultimately IBM Corporation.

Sociology and Sociocybernetics

Systems theory has also been developed within sociology. An important figure in the sociological systems
perspective as developed from GST is Walter Buckley (who from Bertalanffy's theory). Niklas Luhmann (see
Luhmann 1994) is also predominant in the literatures for sociology and systems theory. Miller's living systems
theory was particularly influential in sociology from the time of the early systems movement. Models for dynamic
equilibrium in systems analysis that contrasted classical views from Talcott Parsons and George Homans were
influential in integrating concepts with the general movement. With the renewed interest in systems theory on the
rise since the 1990s, Bailey (1994) notes the concept of systems in sociology dates back to Auguste Comte in the
19th century, Herbert Spencer and Vilfredo Pareto, and that sociology was readying into its centennial as the new
systems theory was emerging following the World Wars. To explore the current inroads of systems theory into
sociology (primarily in the form of complexity science) see sociology and complexity science.

Systems theory 16

In sociology, members of Research Committee 5 1 of the International Sociological Association (which focuses on
sociocybernetics), have sought to identify the sociocybernetic feedback loops which, it is argued, primarily control
the operation of society. On the basis of research largely conducted in the area of education, Raven (1995) has, for
example, argued that it is these sociocybernetic processes which consistently undermine well intentioned public
action and are currently heading our species, at an exponentially increasing rate, toward extinction. See
sustainability. He suggests that an understanding of these systems processes will allow us to generate the kind of
(non "common-sense") targeted interventions that are required for things to be otherwise - i.e. to halt the destruction
of the planet.

Industrial designer, and founder of The Venus Project, Jacque Fresco advocates the utilization of sociocybernetics
for the benefits it could bring to society. A major theme of Fresco's is the concept of a resource-based economy that
replaces the need for the current monetary economy, which is "scarcity-oriented" or "scarcity-based". Fresco argues
that the world is rich in natural resources and energy and that — with modern technology and judicious efficiency —
the needs of the global population can be met with abundance, while at the same time removing the current
limitations of what is deemed possible due to notions of economic viability.

Systems biology

Systems biology is a term used to describe a number of trends in bioscience research, and a movement which draws

on those trends. Proponents describe systems biology as a biology-based inter-disciplinary study field that focuses on

complex interactions in biological systems, claiming that it uses a new perspective (holism instead of reduction).

Particularly from year 2000 onwards, the term is used widely in the biosciences, and in a variety of contexts. An

often stated ambition of systems biology is the modeling and discovery of emergent properties, properties of a

system whose theoretical description is only possible using techniques which fall under the remit of systems biology.

[23]
The term systems biology is thought to have been created by Ludwig von Bertalanffy in 1928.

System dynamics

System Dynamics was founded in the late 1950s by Jay W. Forrester of the MIT Sloan School of Management with
the establishment of the MIT System Dynamics Group. At that time, he began applying what he had learned about
systems during his work in electrical engineering to everyday kinds of systems. Determining the exact date of the
founding of the field of system dynamics is difficult and involves a certain degree of arbitrariness. Jay W. Forrester
joined the faculty of the Sloan School at MIT in 1956, where he then developed what is now System Dynamics. The
first published article by Jay W. Forrester in the Harvard Business Review on "Industrial Dynamics", was published
in 1958. The members of the System Dynamics Society have chosen 1957 to mark the occasion as it is the year in
which the work leading to that article, which described the dynamics of a manufacturing supply chain, was done.

As an aspect of systems theory, system dynamics is a method for understanding the dynamic behavior of complex
systems. The basis of the method is the recognition that the structure of any system — the many circular,
interlocking, sometimes time-delayed relationships among its components — is often just as important in
determining its behavior as the individual components themselves. Examples are chaos theory and social dynamics.
It is also claimed that, because there are often properties-of-the-whole which cannot be found among the
properties-of-the-elements, in some cases the behavior of the whole cannot be explained in terms of the behavior of
the parts. An example is the properties of these letters which when considered together can give rise to meaning
which does not exist in the letters by themselves. This further explains the integration of tools, like language, as a
more parsimonious process in the human application of easiest path adaptability through interconnected systems.

Systems theory 17

Systems engineering

Systems engineering is an interdisciplinary approach and means for enabling the realization and deployment of
successful systems. It can be viewed as the application of engineering techniques to the engineering of systems, as
well as the application of a systems approach to engineering efforts. Systems engineering integrates other
disciplines and specialty groups into a team effort, forming a structured development process that proceeds from

concept to production to operation and disposal. Systems engineering considers both the business and the technical

T251
needs of all customers, with the goal of providing a quality product that meets the user needs.

Systems psychology

Systems psychology is a branch of psychology that studies human behaviour and experience in complex systems. It
is inspired by systems theory and systems thinking, and based on the theoretical work of Roger Barker, Gregory
Bateson, Humberto Maturana and others. It is an approach in psychology, in which groups and individuals, are
considered as systems in homeostasis. Systems psychology "includes the domain of engineering psychology, but in
addition is more concerned with societal systems and with the study of motivational, affective, cognitive and group
behavior than is engineering psychology." In systems psychology "characteristics of organizational behaviour for

example individual needs, rewards, expectations, and attributes of the people interacting with the systems are

T271
considered in the process in order to create an effective system". The Systems psychology includes an illusion of

homeostatic systems, although most of the living systems are in a continuous disequilibrium of various degrees.

References

[I] Bertalanffy (1950: 142)
[2] (Laszlo 1974)

[3] (Schein 1980: 4-11)

[4] Laslo (1972: 14-15)

[5] (Banathy 1997: f 22)

[6] 1968, General System theory: Foundations, Development, Applications, New York: George Braziller, revised edition 1976: ISBN

0-8076-0453-4
[7] (see Steiss 1967; Buckley, 1967)
[8] Peter Senge (2000: 27-49)
[9] (Bailey 1994: 3-8; see also Owens 2004)
[10] (Bailey 1994: 3-8)

[II] (Bailey 1994; Flood 1997; Checkland 1999; Laszlo 1972)

[12] http://www.cscanada.net/index.php/css/article/viewFile/1512/1725

[13] (Hammond 2003: 12-13)

[14] Hammond 2003: 5-9

[15] Hull 1970

[16] (Hammond 2003: 229-233)

[17] Karl Ludwig von Bertalanffy: ... aber vom Menschen wissen wir nichts, (English title: Robots, Men and Minds), translated by Dr.

Hans-Joachim Flechtner. page 115. Econ Verlag GmbH (1970), Dusseldorf, Wien. 1st edition.
[18] (GSTp.32)

[19] perspectives_on_general_system_theory [ProjectsISSS] (http://projects.isss.org/perspectives_on_general_system_theory)
[20] von Bertalanffy, Ludwig, (1974) Perspectives on General System Theory Edited by Edgar Taschdjian. George Braziller, New York
[21] main_systemsinquiry [ProjectsISSS] (http://projects.isss.org/Main/SystemsInquiry)
[22] (see Ash 1992: 198-207)
[23] 1928, Kritische Theorie der Formbildung, Borntraeger. In English: Modern Theories of Development: An Introduction to Theoretical

Biology, Oxford University Press, New York: Harper, 1933
[24] Thome, Bernhard (1993). Systems Engineering: Principles and Practice of Computer-based Systems Engineering. Chichester: John Wiley i

Sons. ISBN 0-471-93552-2.
[25] INCOSE. "What is Systems Engineering" (http://www.incose.org/practice/whatissystemseng.aspx). . Retrieved 2006-1 1-26.
[26] Lester R. Bittel and Muriel Albers Bittel (1978), Encyclopedia of Professional Management, McGraw-Hill, ISBN 0-07-005478-9, p.498.
[27] Michael M. Behrmann (1984), Handbook of Microcomputers in Special Education. College Hill Press. ISBN 0-933014-35-X. Page 212.

Systems theory 18

Further reading

Ackoff, R. (1978). The art of problem solving. New York: Wiley.

Ash, M.G. (1992). "Cultural Contexts and Scientific Change in Psychology: Kurt Lewin in Iowa." American

Psychologist, Vol. 47, No. 2, pp. 198-207.

Bailey, K.D. (1994). Sociology and the New Systems Theory: Toward a Theoretical Synthesis. New York: State

of New York Press.

Banathy, B (1996) Designing Social Systems in a Changing World New York Plenum

Banathy, B. (1991) Systems Design of Education. Englewood Cliffs: Educational Technology Publications

Banathy, B. (1992) A Systems View of Education. Englewood Cliffs: Educational Technology Publications.

ISBN 0-87778-245-8

Banathy, B.H. (1997). "A Taste of Systemics" (http://www.newciv.org/ISSS_Primer/asem04bb.html), The

Primer Project, Retrieved May 14, (2007)

Bateson, G. (1979). Mind and nature: A necessary unity. New York: Ballantine

Bausch, Kenneth C. (2001) The Emerging Consensus in Social Systems Theory, Kluwer Academic New York

ISBN 0-306-46539-6

Ludwig von Bertalanffy (1968). General System Theory: Foundations, Development, Applications New York:

George Braziller

Bertalanffy, L. von (1950), "An Outline of General System Theory" (http://www.isnature.org/events/2009/

Summer/r/Bertalanffyl950-GST_Outline_SELECT.pdf), British Journal for the Philosophy of Science Vol. 1

(No. 2), retrieved 24 October 2010

Bertalanffy, L. von. (1955). "An Essay on the Relativity of Categories." Philosophy of Science, Vol. 22, No. 4,

pp. 243-263.

Bertalanffy, Ludwig von. (1968). Organismic Psychology and Systems Theory. Worchester: Clark University

Press.

Bertalanffy, Ludwig Von. (1974). Perspectives on General System Theory Edited by Edgar Taschdjian. George

Braziller, New York.

Buckley, W. (1967). Sociology and Modern Systems Theory. New Jersey: Englewood Cliffs.

Mario Bunge (1979) Treatise on Basic Philosophy, Volume 4. Ontology II A World of Systems. Dordrecht,

Netherlands: D. Reidel.

Capra, F. (1997). The Web of Life-A New Scientific Understanding of Living Systems, Anchor ISBN

978-0-385-47676-8

Checkland, P. (1981). Systems thinking, Systems practice. New York: Wiley.

Checkland, P. 1997. Systems Thinking, Systems Practice. Chichester: John Wiley & Sons, Ltd.

Churchman, C.W. (1968). The systems approach. New York: Laurel.

Churchman, C.W. (1971). The design of inquiring systems. New York: Basic Books.

Corning, P. (1983) The Synergism Hupothesis: A Theory of Progressive Evolution. New York: McGraw Hill

Davidson, Mark. (1983). Uncommon Sense: The Life and Thought of Ludwig von Bertalanffy, Father of General

Systems Theory. Los Angeles: J. P. Tarcher, Inc.

Durand, D. La systemique, Presses Universitaires de France

Flood, R.L. 1999. Rethinking the Fifth Discipline: Learning within the unknowable." London: Routledge.

Charles Francois. (2004). Encyclopedia of Systems and Cybernetics, Introducing the 2nd Volume (http://

benking.de/systems/encyclopedia/concepts-and-models.htm) and further links to the ENCYCLOPEDIA, K G

Saur, Munich (http://benking.de/encyclopedia/) see also (http://wwwu.uni-klu.ac.at/gossimit/ifsr/francois/

encyclopedia.htm)

Kahn, Herman. (1956). Techniques of System Analysis, Rand Corporation

Laszlo, E. (1995). The Interconnected Universe. New Jersey, World Scientific. ISBN 981-02-2202-5

Systems theory 19

• Francois, C. (1999). Systemics and Cybernetics in a Historical Perspective (http://www.uni-klu.ac.at/
~gossimit/ifsr/francois/papers/systemics_and_cybernetics_in_a_historical_perspective.pdf)
Jantsch, E. (1980). The Self Organizing Universe. New York: Pergamon.

Gorelik, G. (1975) Reemergence of Bogdanov's Tektology in. Soviet Studies of Organization, Academy of
Management Journal. 18/2, pp. 345—357

Hammond, D. 2003. The Science of Synthesis. Colorado: University of Colorado Press.
Hinrichsen, D. and Pritchard, A.J. (2005) Mathematical Systems Theory. New York: Springer. ISBN
978-3-540-44125-0

Hull, D.L. 1970. "Systemic Dynamic Social Theory." Sociological Quarterly, Vol. 11, Issue 3, pp. 351—363.
Hyotyniemi, H. (2006). T^eo cybernetics in Biological Systems (http://neocybernetics.com/reportl51/). Espoo:
Helsinki University of Technology, Control Engineering Laboratory.
Jackson, M.C. 2000. Systems Approaches to Management. London: Springer.

Klir, G.J. 1969. An Approach to General Systems Theory. New York: Van Nostrand Reinhold Company.
Ervin Laszlo 1972. The Systems View of the World. New York: George Brazilier.

Laszlo, E. (1972a). The systems view of the world. The natural philosophy of the new developments in the
sciences. New York: George Brazillier. ISBN 0-8076-0636-7

Laszlo, E. (1972b). Introduction to systems philosophy. Toward a new paradigm of contemporary thought. San
Francisco: Harper.

Laszlo, Ervin. 1996. The Systems View of the World. Hampton Press, NJ. (ISBN 1-57273-053-6).
Lemkow, A. (1995) The Wholeness Principle: Dynamics of Unity Within Science, Religion & Society. Quest
Books, Wheaton.

Niklas Luhmann (1996),"Social Systems",Stanford University Press, Palo Alto, CA

Mattessich, R. (1978) Instrumental Reasoning and Systems Methodology: An Epistemology of the Applied and
Social Sciences. Reidel, Boston

Minati, Gianfranco. Collen, Arne. (1997) Introduction to Systemics Eagleye books. ISBN 0-924025-06-9
Montuori, A. (1989). Evolutionary Competence. Creating the Future. Amsterdam: Gieben.
Morin, E. (2008). On Complexity. Cresskill, NJ: Hampton Press.

Odum, H. (1994) Ecological and General Systems: An introduction to systems ecology, Colorado University
Press, Colorado.

Olmeda, Christopher J. (1998). Health Informatics: Concepts of Information Technology in Health and Human
Services. Delfin Press. ISBN 0-9821442-1-0

Owens, R.G. (2004). Organizational Behavior in Education: Adaptive Leadership and School Reform, Eighth
Edition. Boston: Pearson Education, Inc.

Pharaoh, M.C. (online). Looking to systems theory for a reductive explanation of phenomenal experience and
evolutionary foundations for higher order thought (http://homepage.ntlworld.eom/m.pharoah/) Retrieved
Dec. 14 2007.

Science as Paradigmatic Complexity by Wallace H. Provost Jr. (http://philosophy.freeopenu.org/mod/
resource/view. php?id=8721) 1984 in the International Journal of General Systems
Schein, E.H. (1980). Organizational Psychology, Third Edition. New Jersey: Prentice-Hall.
Peter Senge (1990). The Fifth Discipline. The art and practice of the learning organization. New York:
Doubleday.

Senge, P., Ed. (2000). Schools That Learn: A Fifth Discipline Fieldbook for Educators, Parents, and Everyone
Who Cares About Education. New York: Doubleday Dell Publishing Group.

Snooks, GD. (2008). "A general theory of complex living systems: Exploring the demand side of dynamics",
Complexity, 13: 12-20.
Steiss, A.W. (1967). Urban Systems Dynamics. Toronto: Lexington Books.

Systems theory 20

• Gerald Weinberg. (1975). An Introduction to General Systems Thinking (1975 ed., Wiley-Interscience) (2001 ed.
Dorset House).

• Wiener, N. (1967). The human use of human beings. Cybernetics and Society. New York: Avon.

• Young, O. R., "A Survey of General Systems Theory", General Systems, vol. 9 (1964), pages 61—80. (overview
about different trends and tendencies, with bibliography)

External links

• Systems theory (http://pespmcl.vub.ac.be/SYSTHEOR.html) at Principia Cybernetica Web
Organizations

• International Society for the System Sciences (http://projects.isss.org/Main/Primer)

• New England Complex Systems Institute (http://www.necsi.edu/)

• System Dynamics Society (http://www.systemdynamics.org/)

Systems analysis

Systems analysis is the study of sets of interacting entities, including computer systems analysis. This field is
closely related to requirements analysis or operations research. It is also "an explicit formal inquiry carried out to
help someone (referred to as the decision maker) identify a better course of action and make a better decision than he
might otherwise have made."

Overview

The terms analysis and synthesis come from Greek where they mean respectively "to take apart" and "to put
together". These terms are in scientific disciplines from mathematics and logic to economy and psychology to denote
similar investigative procedures. Analysis is defined as the procedure by which we break down an intellectual or
substantial whole into parts. Synthesis is defined as the procedure by which we combine separate elements or
components in order to form a coherent whole. Systems analysis researchers apply methodology to the analysis of
systems involved to form an overall picture. System analysis is used in every field where there is a work of
developing something.

Information technology

The development of a computer-based information system includes a systems analysis phase which produces or
enhances the data model which itself is a precursor to creating or enhancing a database (see Christopher J. Date "An
Introduction to Database Systems"). There are a number of different approaches to system analysis. When a
computer-based information system is developed, systems analysis (according to the Waterfall model) would
constitute the following steps:

• The development of a feasibility study, involving determining whether a project is economically, socially,
technologically and organizationally feasible.

• Conducting fact-finding measures, designed to ascertain the requirements of the system's end-users. These
typically span interviews, questionnaires, or visual observations of work on the existing system.

• Gauging how the end-users would operate the system (in terms of general experience in using computer hardware
or software), what the system would be used for etc.

Another view outlines a phased approach to the process. This approach breaks systems analysis into 5 phases:

• Scope definition

• Problem analysis

Systems analysis 21

• Requirements analysis

• Logical design

• Decision analysis

Use cases are a widely-used systems analysis modeling tool for identifying and expressing the functional
requirements of a system. Each use case is a business scenario or event for which the system must provide a defined
response. Use cases evolved out of object-oriented analysis; however, their use as a modeling tool has become
common in many other methodologies for system analysis and design.

Practitioners

Practitioners of systems analysis are often called up to dissect systems that have grown haphazardly to determine the
current components of the system. This was shown during the year 2000 re-engineering effort as business and
manufacturing processes were examined as part of the Y2K automation upgrades. Employment utilizing systems
analysis include systems analyst, business analyst, manufacturing engineer, enterprise architect, etc.

While practitioners of systems analysis can be called upon to create new systems, they often modify, expand or
document existing systems (processes, procedures and methods). A set of components interact with each other to
accomplish some specific purpose. Systems are all around us. Our body is itself a system. A business is also a
system. People, money, machine, market and material are the components of business system that work together that
achieve the common goal of the organization.

References

[1] SYSTEMS ANALYSIS (http://web.archive.org/web/20070822025602/http://pespmcl.vub.ac.be/ASC/SYSTEM_ANALY.html)
[2] Tom Ritchey, [http://www.swemorph.com/pdf/anaeng-r.pdfAnalysis and .

External links

• Systems Analysis, Modelling and Prediction (SAMP), University of Oxford (http://www.eng.ox.ac.uk/samp)

• Software Requirement Analysis using UML (http://www.slideshare.net/dhirajmusings/
software-requirement-analysis-using-uml) article by Dhiraj Shetty.

• Introduction to Social Macrodynamics (http://urss.ru/cgi-bin/db. pl?cp=&page=Book&id=34250&lang=en&
blang=en&list=Found)

• A useful set of guides and a case study about the practical application of business and systems analysis methods
(http://www.cilco.co.uk/briefing-studies/index.html)

• Complete online tutorial for system analysis and design (http://www.systemsanalysis.co.nr)

• A comprehensive description of the discipline of systems analysis from Simmons College, Boston, MA, USA
(www.simmons.edu) (http://web.simmons.edu/~benoit/LIS486/SystemsAnalysis.html)

Relational theory

In physics and philosophy, a relational theory is a framework to understand reality or a physical system in such a
way that the positions and other properties of objects are only meaningful relative to other objects. In a relational
spacetime theory, space does not exist unless there are objects in it; nor does time exist without events. The relational
view proposes that space is contained in objects and that an object represents within itself relationships to other
objects. Space can be defined through the relations among the objects that it contains considering their variations
through time. The alternative spatial theory is an absolute theory in which the space exists independently of any
objects that can be immersed in it.

Someone who has constructed or a relational theory or promotes relational theorising is called a relationist.

The relational point of view was advocated by in physics by Gottfried von Leibniz, Ernst Mach (in his Mach's
principle), and it was rejected by Isaac Newton in his successful description of classical physics. Although Albert
Einstein was impressed by Mach's principle, he did not fully incorporate it into his theory of general relativity.
Several attempts have been made to formulate a full Machian theory, but most physicists think that none have so far
succeeded. For example, see Brans-Dicke theory

A Relational approach to quantum physics has been developed, in analogy with Einstein's special relativity of space
and time.

Relationist physicists such as John Baez and Carlo Rovelli have criticised the leading unified theory of gravity and
quantum mechanics, string theory, as retaining absolute space. Some prefer a developing theory of gravity, loop
quantum gravity for its 'backgroundlessness'.

Differential Equations

A differential equation is a

mathematical equation for an unknown
function of one or several variables
that relates the values of the function
itself and its derivatives of various
orders. Differential equations play a
prominent role in engineering, physics,
economics, and other disciplines.

Differential equations arise in many
areas of science and technology,
specifically, whenever a deterministic
relation involving some continuously
varying quantities (modeled by
functions) and their rates of change in
space and/or time (expressed as
derivatives) is known or postulated.
This is illustrated in classical
mechanics, where the motion of a body
is described by its position and

Visualization of heat transfer in a pump casing, by solving the heat equation. Heat is

being generated internally in the casing and being cooled at the boundary, providing a

steady state temperature distribution.

velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces
acting on the body and state this relation as a differential equation for the unknown position of the body as a function

Differential Equations 23

of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential equations is determination of the velocity of a ball
falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the
acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be
modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its
velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their
solutions — the set of functions that satisfy the equation. Only the simplest differential equations admit solutions
given by explicit formulas; however, some properties of solutions of a given differential equation may be determined
without finding their exact form. If a self-contained formula for the solution is not available, the solution may be
numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis
of systems described by differential equations, while many numerical methods have been developed to determine
solutions with a given degree of accuracy.

Directions of study

The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and
engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure
mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the
rigorous justification of the methods for approximating solutions. Differential equations play an important role in
modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to
interactions between neurons. Differential equations such as those used to solve real-life problems may not
necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using
numerical methods.

Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not

have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in

more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic

type.

The study of the stability of solutions of differential equations is known as stability theory.

Nomenclature

The theory of differential equations is quite developed and the methods used to study them vary significantly with
the type of the equation.

• An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as
the dependent variable) is a function of a single independent variable. In the simplest form, the unknown
function is a real or complex valued function, but more generally, it may be vector-valued or matrix- valued: this
corresponds to considering a system of ordinary differential equations for a single function. Ordinary differential
equations are further classified according to the order of the highest derivative of the dependent variable with
respect to the independent variable appearing in the equation. The most important cases for applications are
first-order and second-order differential equations. In the classical literature also distinction is made between
differential equations explicitly solved with respect to the highest derivative and differential equations in an
implicit form.

• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of
multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to
the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic
equations, especially for second-order linear equations, is of utmost importance. Some partial differential

Differential Equations 24

equations do not fall into any of these categories over the whole domain of the independent variables and they are
said to be of mixed type.

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential
equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and
nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of
an appropriate function space, which results in much more developed theory of linear differential equations.
Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear
subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the
unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the
independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear
differential equation.

There are very few methods of explicitly solving nonlinear differential equations; those that are known typically
depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated
behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence,
uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and
boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to
be a significant advance in the mathematical theory (cf. Navier— Stokes existence and smoothness).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are
only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the
nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Examples

In the first group of examples, let u be an unknown function of x, and c and w are known constants.

• Inhomogeneous first-order linear constant coefficient ordinary differential equation:

du 2

— = CU + X .

• Homogeneous second-order linear ordinary differential equation:

d 2 u du

dx 2 dx

• Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic

oscillator:

d*u

dx 2

• First-order nonlinear ordinary differential equation:

— = u 2 + l.
ax

• Second-order nonlinear ordinary differential equation describing the motion of a pendulum of length L:

T d 2 u

L—— + g sin u — 0.
dx 1

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

• Homogeneous first-order linear partial differential equation:

du du

dt dx

• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace

equation:

+ UJ 2 U = 0.

Differential Equations 25

d 2 u d 2 u
dx 2 dy 2

• Third-order nonlinear partial differential equation, the Korteweg— de Vries equation:

du du d 3 u

dt dx dx 3

Related concepts

• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in
which the derivative of the function at a certain time is given in terms of the values of the function at earlier
times.

• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and
the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion
equations.

• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms,
given in implicit form.

Connection to difference equations

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates
assume only discrete values, and the relationship involves values of the unknown function or functions and values at
nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties
of differential equations involve approximation of the solution of a differential equation by the solution of a
corresponding difference equation.

Universality of mathematical description

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and
economics differential equations are used to model the behavior of complex systems. The mathematical theory of
differential equations first developed, together with the sciences, where the equations had originated and where the
results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may
give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be
viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in
the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order
partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much
like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is
governed by another second-order partial differential equation, the heat equation. It turned out that many diffusion
processes, while seemingly different, are described by the same equation; Black-Scholes equation in finance is for
instance, related to the heat equation.

Differential Equations

Exact solutions

Some differential equations have solutions which can be written in an exact and closed form. These are given here.

In the table below, H(x), Z(x) or H(y) and Z(y) are any integrable functions of x or y, and A, B, C, I, L, N, M are all
constants. In general A, B, C, I, L, are real numbers, but N, M, P and Q may be complex. The differential equations
are in their equivalent and alternative forms which lead to the solution through integration.

Differential Equation

General Solution

dy = F(x)dx

y = F(x)dx

dy = F(y)dx

f dy
X ~J F{y)

H(y)^ + Z(x)=0
H(y)dy + Z(x)dx =

J H(y)dy + J Z(x)dx = C

^ + H{x)y + Z(x) =
da;

dy + H(x)ydx + Z{x)dx =

y =- e -jm*)** f e f H M d *Z(x)dx

S = ™

r - i f dy | r,

J ^2jF(y)dy + C 1

d» du
dar dx

If I 2 > AL

then y = Ne {-'+^-*m + Me-V^ 1 *-"-)*

If I 2 = AL

then y = (Ax + B)e- Ix l 2

If I 2 < AL

then y = e~ 2

Psinf v /|7 2 -4L|^J+Qco S

{Ir-

-<)]

Notable differential equations

Newton's Second Law in dynamics (mechanics)

Hamilton's equations in classical mechanics

Radioactive decay in nuclear physics

Newton's law of cooling in thermodynamics

The wave equation

Maxwell's equations in electromagnetism

The heat equation in thermodynamics

Laplace's equation, which defines harmonic functions

Poisson's equation

Einstein's field equation in general relativity

The Schrodinger equation in quantum mechanics

The geodesic equation

The Navier— Stokes equations in fluid dynamics

The Cauchy— Riemann equations in complex analysis

The Poisson— Boltzmann equation in molecular dynamics

The shallow water equations

Differential Equations 27

• Universal differential equation

• The Lorenz equations whose solutions exhibit chaotic flow.

Biology

• Verhulst equation — biological population growth

• von Bertalanffy model — biological individual growth

• Lotka— Volterra equations — biological population dynamics

• Replicator dynamics — may be found in theoretical biology

• Hodgkin-Huxley model - neural action potentials

Economics

• The Black-Scholes PDE

• Exogenous growth model

• Malthusian growth model

• The Vidale-Wolfe advertising model

References

• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.

• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition),
Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.

• W. Johnson, A Treatise on Ordinary and Partial Differential Equations , John Wiley and Sons, 1913, in
University of Michigan Historical Math Collection

• E.L. Ince, Ordinary Differential Equations, Dover Publications, 1956

• E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955

• P. Blanchard, R.L. Devaney, G.R. Hall, Differential Equations, Thompson, 2006

• Calculus, Teach Yourself, P.Abbott and H. Neill, 2003 pages 266-277

• Further Elementary Analysis, R.I.Porter, 1978, chapter XIX Differential Equations</ref>

External links

Lectures on Differential Equations MIT Open Courseware Videos

[41
Online Notes / Differential Equations Paul Dawkins, Lamar University

Differential Equations , S.O.S. Mathematics

Introduction to modeling via differential equations Introduction to modeling by means of differential

equations, with critical remarks.

[71
Differential Equation Solver Java applet tool used to solve differential equations.

[O]

Mathematical Assistant on Web Symbolic ODE tool, using Maxima

[Ql

Exact Solutions of Ordinary Differential Equations

Collection of ODE and DAE models of physical systems MATLAB models

Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by

JiriLeblofUIUC

Differential Equations 28

References

[I] http://www.hti.umich.edu/cgi/b/bib/bibperm?ql=abv5010.0001.001
[2] http://hti.umich.edU/u/umhistmath/

[3] http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/

[4] http://tutorial.math.lamar.edu/classes/de/de.aspx

[5] http://www.sosmath.com/diffeq/diffeq.html

[6] http://www.diptem.unige.it/patrone/differential_equations_intro.htm

[7] http://publicliterature.org/tools/differential_equation_solver/

[8] http://user.mendelu.cz/marik/maw/index. php?lang=en&form=ode

[9] http://eqworld.ipmnet.ru/en/solutions/ode.htm

[10] http://www.hedengren.net/research/models.htm

[II] http://www.jirka.org/diffyqs/

Computational physics

Computational physics is the study and implementation of numerical algorithms to solve problems in physics for
which a quantitative theory already exists. It is often regarded as a subdiscipline of theoretical physics but some
consider it an intermediate branch between theoretical and experimental physics.

Physicists often have a very precise mathematical theory describing how a system will behave. Unfortunately, it is
often the case that solving the theory's equations ab initio in order to produce a useful prediction is not practical. This
is especially true with quantum mechanics, where only a handful of simple models admit closed-form, analytic
solutions. In cases where the equations can only be solved approximately, computational methods are often used.

Applications of computational physics

Computation now represents an essential component of modern research in accelerator physics, astrophysics, fluid
mechanics, lattice field theory/lattice gauge theory (especially lattice quantum chromodynamics), plasma physics
(see plasma modeling), solid state physics and soft condensed matter physics. Computational solid state physics, for
example, uses density functional theory to calculate properties of solids, a method similar to that used by chemists to
study molecules.

As these topics are explored, many more general numerical and mathematical problems are encountered in order to
calculate physical properties of the modeled systems. These include, but are not limited to

• Solving differential equations

• Evaluating integrals

• Stochastic methods, especially Monte Carlo methods

• Specialized partial differential equation methods, for example the finite difference method and the finite element
method

• The matrix eigenvalue problem — the problem of finding eigenvalues of very large matrices, and their
corresponding eigenvectors (eigenstates in quantum physics)

• The pseudo-spectral method

Computational physics also encompasses the tuning of the software/hardware structure to solve problems.
Approaches to solving the problems are often very demanding in terms of processing power and/or memory requests.

Computational physics 29

External links

• C20 IUPAP Commission on Computational Physics

• APSDCOMP [2]

• IoP CPG (UK) [3]

• SciDAC: Scientific Discovery through Advanced Computing

• Open Source Physics

• SCINET Scientific Software Framework

References

[1] http

[2] http

[3] http

[4] http

[5] http

[6] http

//phycomp. technion.ac.il/~C20

//www. aps.org/units/dcomp/index.cfm

//www. iop.org/activity/groups/subject/comp/index. html

//www. scidac.gov/physics/physics. html

//www. compadre.org/OSP/

//www. ohacs.com

Dynamical Systems and Symbolic Dynamics
System

System (from Latin systema, in turn from Greek
avaTrjjia systema, "whole compounded of several parts
or members, system", literary "composition" ) is a set
of interacting or interdependent system components
forming an integrated whole.

The concept of an "integrated whole" can also be
stated in terms of a system embodying a set of
relationships which are differentiated from
relationships of the set to other elements, and from
relationships between an element of the set and
elements not a part of the relational regime.

SURROUNDINGS

A-

SYSTEM

^•BOUNDARY

The scientific research field which is engaged in the
study of the general properties of systems include

systems theory, cybernetics, dynamical systems, A schematic representation of a closed system and its boundary

thermodynamics and complex systems. They

investigate the abstract properties of the matter and organization, searching concepts and principles which are

independent of the specific domain, substance, type, or temporal scales of existence.

Most systems share common characteristics, including:

• Systems have structure, defined by components and their composition;

• Systems have behavior, which involves inputs, processing and outputs of material, energy, information, or data;

• Systems have interconnect! vity: the various parts of a system have functional as well as structural relationships
between each other.

• Systems may have some functions or groups of functions

The term system may also refer to a set of rules that governs structure and/or behavior.

History

The word system in its meaning here, has a long history which can be traced back to Plato (Philebus), Aristotle
(Politics) and Euclid (Elements). It had meant "total", "crowd" or "union" in even more ancient times, as it derives
from the verb sunistemi, uniting, putting together.

In the 19th century the first who developed the concept of a "system" in the natural sciences was the French physicist
Nicolas Leonard Sadi Carnot who studied thermodynamics. In 1824 he studied the system which he called the
working substance, i.e. typically a body of water vapor, in steam engines, in regards to the system's ability to do
work when heat is applied to it. The working substance could be put in contact with either a boiler, a cold reservoir
(a stream of cold water), or a piston (to which the working body could do work by pushing on it). In 1850, the
German physicist Rudolf Clausius generalized this picture to include the concept of the surroundings and began to
use the term "working body" when referring to the system.

One of the pioneers of the general systems theory was the biologist Ludwig von Bertalanffy. In 1945 he introduced
models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular

System 32

kind, the nature of their component elements, and the relation or 'forces' between them.

Significant development to the concept of a system was done by Norbert Wiener and Ross Ashby who pioneered the
use of mathematics to study systems.

In the 1980s the term complex adaptive system was coined at the interdisciplinary Santa Fe Institute by John H.
Holland, Murray Gell-Mann and others.

System concepts

Environment and boundaries

Systems theory views the world as a complex system of interconnected parts. We scope a system by defining
its boundary; this means choosing which entities are inside the system and which are outside - part of the
environment. We then make simplified representations (models) of the system in order to understand it and to
predict or impact its future behavior. These models may define the structure and/or the behavior of the system.

Natural and man-made systems

There are natural and man-made (designed) systems. Natural systems may not have an apparent objective but
their outputs can be interpreted as purposes. Man-made systems are made with purposes that are achieved by
the delivery of outputs. Their parts must be related; they must be "designed to work as a coherent entity" - else
they would be two or more distinct systems.

Theoretical Framework

An open system exchanges matter and energy with its surroundings. Most systems are open systems; like a
car, coffeemaker, or computer. A closed system exchanges energy, but not matter, with its environment; like
Earth or the project Biosphere2 or 3. An isolated system exchanges neither matter nor energy with its
environment. A theoretical example of such system is the Universe.

Process and transformation process

A system can also be viewed as a bounded transformation process, that is, a process or collection of processes
that transforms inputs into outputs. Inputs are consumed; outputs are produced. The concept of input and
output here is very broad. E.g., an output of a passenger ship is the movement of people from departure to
destination.

Subsystem

A subsystem is a set of elements, which is a system itself, and a component of a larger system.

System Model

A system comprises multiple views. For the man-made systems it may be such views as planning,
requirement, design, implementation, deployment, operational, structure, behavior, input data, and output data
views. A system model is required to describe and represent all these multiple views.

System Architecture

A system architecture, using one single coalescence model for the description of multiple views such as
planning, requirement, design, implementation, deployment, operational, structure, behavior, input data, and
output data views, is a kind of system model.

System 33

Types of systems

Evidently, there are many types of systems that can be analyzed both quantitatively and qualitatively. For example,
with an analysis of urban systems dynamics, [A.W. Steiss] defines five intersecting systems, including the
physical subsystem and behavioral system. For sociological models influenced by systems theory, where Kenneth D.
Bailey defines systems in terms of conceptual, concrete and abstract systems; either isolated, closed, or open,
Walter F. Buckley defines social systems in sociology in terms of mechanical, organic, and process models. Bela

ro]

H. Banathy cautions that with any inquiry into a system that understanding the type of system is crucial and
defines Natural and Designed systems.

In offering these more global definitions, the author maintains that it is important not to confuse one for the other.
The theorist explains that natural systems include sub-atomic systems, living systems, the solar system, the galactic
system and the Universe. Designed systems are our creations, our physical structures, hybrid systems which include
natural and designed systems, and our conceptual knowledge. The human element of organization and activities are
emphasized with their relevant abstract systems and representations. A key consideration in making distinctions
among various types of systems is to determine how much freedom the system has to select purpose, goals, methods,
tools, etc. and how widely is the freedom to select itself distributed (or concentrated) in the system.

George J. Klir maintains that no "classification is complete and perfect for all purposes," and defines systems in
terms of abstract, real, and conceptual physical systems, bounded and unbounded systems, discrete to continuous,
pulse to hybrid systems, et cetera. The interaction between systems and their environments are categorized in terms
of relatively closed, and open systems. It seems most unlikely that an absolutely closed system can exist or, if it did,
that it could be known by us. Important distinctions have also been made between hard and soft systems. Hard
systems are associated with areas such as systems engineering, operations research and quantitative systems analysis.
Soft systems are commonly associated with concepts developed by Peter Checkland and Brian Wilson through Soft
Systems Methodology (SSM) involving methods such as action research and emphasizing participatory designs.
Where hard systems might be identified as more "scientific," the distinction between them is actually often hard to
define.

Cultural system

A cultural system may be defined as the interaction of different elements of culture. While a cultural system is quite
different from a social system, sometimes both systems together are referred to as the sociocultural system. A major
concern in the social sciences is the problem of order. One way that social order has been theorized is according to
the degree of integration of cultural and social factors.

Economic system

An economic system is a mechanism (social institution) which deals with the production, distribution and
consumption of goods and services in a particular society. The economic system is composed of people, institutions
and their relationships to resources, such as the convention of property. It addresses the problems of economics, like
the allocation and scarcity of resources.

System 34

Application of the system concept

Systems modeling is generally a basic principle in engineering and in social sciences. The system is the
representation of the entities under concern. Hence inclusion to or exclusion from system context is dependent of the
intention of the modeler.

No model of a system will include all features of the real system of concern, and no model of a system must include
all entities belonging to a real system of concern.

Systems in information and computer science

In computer science and information science, system is a software system which has components as its structure and
observable Inter-process communications as its behavior. Again, an example will illustrate: There are systems of
counting, as with Roman numerals, and various systems for filing papers, or catalogues, and various library systems,
of which the Dewey Decimal System is an example. This still fits with the definition of components which are
connected together (in this case in order to facilitate the flow of information).

System can also be used referring to a framework, be it software or hardware, designed to allow software programs
to run, see platform.

Systems in engineering and physics

In engineering and physics, a physical system is the portion of the universe that is being studied (of which a
thermodynamic system is one major example). Engineering also has the concept of a system that refers to all of the
parts and interactions between parts of a complex project. Systems engineering refers to the branch of engineering
that studies how this type of system should be planned, designed, implemented, built, and maintained.

Systems in social and cognitive sciences and management research

Social and cognitive sciences recognize systems in human person models and in human societies. They include
human brain functions and human mental processes as well as normative ethics systems and social/cultural
behavioral patterns.

In management science, operations research and organizational development (OD), human organizations are viewed
as systems (conceptual systems) of interacting components such as subsystems or system aggregates, which are
carriers of numerous complex business processes (organizational behaviors) and organizational structures.
Organizational development theorist Peter Senge developed the notion of organizations as systems in his book The
Fifth Discipline.

Systems thinking is a style of thinking/reasoning and problem solving. It starts from the recognition of system
properties in a given problem. It can be a leadership competency. Some people can think globally while acting
locally. Such people consider the potential consequences of their decisions on other parts of larger systems. This is
also a basis of systemic coaching in psychology.

Organizational theorists such as Margaret Wheatley have also described the workings of organizational systems in
new metaphoric contexts, such as quantum physics, chaos theory, and the self-organization of systems.

Systems applied to strategic thinking

In 1988, military strategist, John A. Warden III introduced his Five Ring System model in his book, The Air
Campaign contending that any complex system could be broken down into five concentric rings. Each
ring — Leadership, Processes, Infrastructure, Population and Action Units — could be used to isolate key elements of
any system that needed change. The model was used effectively by Air Force planners in the First Gulf War.
In the late 1990s, Warden applied this five ring model to business strategy.

System 35

References

[I] avaTr\\.ia (http://www. perseus. tufts. edu/hopper/text?doc=Perseus:text: 1999. 04. 0057:entry=su/sthma), Henry George Liddell, Robert
Scott, A Greek-English Lexicon, on Perseus Digital Library

[2] 1945, Zu einer allgemeinen Systemlehre, Blatter filr deutsche Philosophie, 3/4. (Extract in: Biologia Generalis, 19 (1949), 139-164.
[3] 1948, Cybernetics: Or the Control and Communication in the Animal and the Machine. Paris, France: Librairie Hermann & Cie, and

Cambridge, MA: MIT Press.Cambridge, MA: MIT Press.
[4] 1956. An Introduction to Cybernetics (http://pespmcl.vub.ac.be/ASHBBOOK.html), Chapman & Hall.
[5] Steiss 1967, p.8-18.
[6] Bailey, 1994.
[7] Buckley, 1967.
[8] Banathy, 1997.
[9] Klir 1969, pp. 69-72
[10] Checkland 1997; Flood 1999.

[II] Warden, John A. Ill (1988). The Air Campaign: Planning for Combat. Washington, D.C.: National Defense University Press.
ISBN 9781583481004.

[12] Warden, John A. Ill (September 1995). "Chapter 4: Air theory for the 21st century" (http://www.airpower.maxwell.af.mil/airchronicles/

battle/chp4.html) (in Air and Space Power Journal). Battlefield of the Future: 2 1st Century Warfare Issues. United States Air Force. .

Retrieved December 26, 2008.
[13] Warden, John A. Ill (1995). "Enemy as a System" (http://www.airpower.maxwell.af.mil/airchronicles/apj/apj95/spr95_files/warden.

htm). Airpower Journal Spring (9): 40-55. . Retrieved 2009-03-25.
[14] Russell, Leland A.; Warden, John A. (2001). Winning in FastTime: Harness the Competitive Advantage of Prometheus in Business and in

Life. Newport Beach, CA: GEO Group Press. ISBN 0971269718.

Further reading

• Alexander Backlund (2000). "The definition of system". In: Kybernetes Vol. 29 nr. 4, pp. 444—451.

• Kenneth D. Bailey (1994). Sociology and the New Systems Theory: Toward a Theoretical Synthesis. New York:
State of New York Press.

• Bela H. Banathy (1997). "A Taste of Systemics" (http://www.newciv.org/ISSS_Primer/asem04bb.html), ISSS
The Primer Project.

• Walter F. Buckley (1967). Sociology and Modern Systems Theory, New Jersey: Englewood Cliffs.

• Peter Checkland (1997). Systems Thinking, Systems Practice. Chichester: John Wiley & Sons, Ltd.

• Robert L. Flood (1999). Rethinking the Fifth Discipline: Learning within the unknowable. London: Routledge.

• George J. Klir (1969). Approach to General Systems Theory, 1969.

• Brian Wilson (1980). Systems: Concepts, methodologies and Applications , John Wiley

• Brian Wilson (2001). Soft Systems Methodology — Conceptual model building and its contribution, J.H.Wiley.

• Beynon-Davies P. (2009). Business Information + Systems. Palgrave, Basingstoke. ISBN 978-0-230-20368-6

External links

• Definitions of Systems and Models (http://www.physicalgeography.net/fundamentals/4b.html) by Michael
Pidwirny, 1999-2007.

• Definitionen von "System" (1572-2002) (http://www.muellerscience.com/SPEZIALITAETEN/System/
System_Definitionen.htm) by Roland Muller, 2001-2007 (most in German).

• Theory and Practical Exercises of System Dynamics (http://www.dinamica-de-sistemas.com/libros/dynamics.
htm) by Juan Martin (also in Spanish)

Dynamics 36

Dynamics

Dynamics (from Greek Svva^iKog - dynamikos "powerful", from Svvafug - dynamis "power") may refer to:

Physics and engineering

• Dynamics (mechanics), the time evolution of physical processes

• Aerodynamics, the study of gases in motion

• Analytical dynamics refers to the motion of bodies as induced by external forces

• Flight dynamics, the science of aircraft and spacecraft design

• Fluid dynamics or hydrodynamics, the study of fluid flow

• Computational fluid dynamics, a way of studying fluid dynamics using numerical methods

• Molecular dynamics, the study of motion on the molecular level

• Langevin dynamics, an approach to the mathematical modeling of molecular dynamics

• Brownian dynamics, a simplified version of Langevin dynamics

• Quantum chromodynamics, a theory of the strong interaction (color force)

• Quantum electrodynamics, description of how matter and light interact

• Relativistic dynamics, a combination of relativistic and quantum concepts

• Stellar dynamics, in astrophysics, a description of the collective motion of stars

• System dynamics, the study of the behavior of complex systems

• Thermodynamics, the study of the relationships between heat and mechanical energy

Sociology and psychology

• Group dynamics, the study of social group processes

• Power dynamics, the dynamics of power, used in sociology

• Psychodynamics, the study of the underlying psychological forces driving human behaviour

• Spiral Dynamics, a social development theory

• Social dynamics, the ability of a society to react to changes

Computer science and mathematics

• Dynamic data structure, a data structure where the data elements may change

• Dynamical system, a concept describing a point's time dependency

• Symbolic dynamics, a method to model dynamical systems

• Dynamic programming, a method of solving complex problems by breaking them down into simpler steps

• Dynamic program analysis, a set of methods for analyzing computer software

Dynamics 37

Companies

• Arrow Dynamics, roller coaster designer

• Boston Dynamics, robot designer

• Crystal Dynamics, video game developer

• General Dynamics, defence contractor

Other

• Dynamics (music), the softness or loudness of a sound or note

• Force Dynamics, a semantic concept about how entities interact with reference to force

• Microsoft Dynamics, a line of business software

• Neural oscillation in the field of neurodynamics, a rhythmic pattern in the brain

• Population dynamics, in life sciences, the changes in the composition of a population

Dynamical systems theory

Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical
systems, usually by employing differential equations or difference equations. When differential equations are
employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is
called discrete dynamical systems. When the time variable runs over a set which is discrete over some intervals and
continuous over other intervals or is any arbitrary time-set such as a cantor set then one gets dynamic equations on
time scales. Some situations may also be modelled by mixed operators such as differential-difference equations.

This theory deals with the long-term qualitative behavior of dynamical systems, and the studies of the solutions to
the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary
orbits as well as the behaviour of electronic circuits and the solutions to partial differential equations that arise in
biology. Much of modern research is focused on the study of chaotic systems.

This field of study is also called just Dynamical systems, Systems theory or longer as Mathematical Dynamical
Systems Theory and the Mathematical theory of dynamical systems.

Dynamical systems theory

Overview

Dynamical systems theory and chaos theory deal with the
long-term qualitative behavior of dynamical systems.
Here, the focus is not on finding precise solutions to the
equations defining the dynamical system (which is often
hopeless), but rather to answer questions like "Will the
system settle down to a steady state in the long term, and
if so, what are the possible steady states?", or "Does the
long-term behavior of the system depend on its initial
condition?"

An important goal is to describe the fixed points, or
steady states of a given dynamical system; these are
values of the variable which won't change over time.
Some of these fixed points are attractive, meaning that if
the system starts out in a nearby state, it will converge
towards the fixed point.

The Lorenz attractor is an example of a non-linear dynamical
system. Studying this system helped give rise to Chaos theory.

Similarly, one is interested in periodic points, states of the system which repeat themselves after several timesteps.
Periodic points can also be attractive. Sarkovskii's theorem is an interesting statement about the number of periodic
points of a one-dimensional discrete dynamical system.

Even simple nonlinear dynamical systems often exhibit almost random, completely unpredictable behavior that has
been called chaos. The branch of dynamical systems which deals with the clean definition and investigation of chaos
is called chaos theory.

History

The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences
and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the
state of the system only a short time into the future.

Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical
techniques and could only be accomplished for a small class of dynamical systems.

Some excellent presentations of mathematical dynamic system theory include Beltrami (1987), Luenberger (1979),

Padulo and Arbib (1974), and Strogatz (1994)

[l]

Concepts
Dynamical systems

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time
dependence of a point's position in its ambient space. Examples include the mathematical models that describe the
swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an
appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The
numbers are also the coordinates of a geometrical space — a manifold. The evolution rule of the dynamical system is
a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time
interval only one future state follows from the current state.

Dynamical systems theory 39

Dynamicism

Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic
cognition, is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues
that differential equations are more suited to modelling cognition than more traditional computer models.

Nonlinear system

In mathematics, a nonlinear system is a system which is not linear, i.e. a system which does not satisfy the
superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to be solved for
cannot be written as a linear sum of independent components. A nonhomogenous system, which is linear apart from
the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems
are usually studied alongside linear systems, because they can be transformed to a linear system as long as a
particular solution is known.

Related fields
Arithmetic dynamics

Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics,
dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of
self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic
properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or
rational function.

Chaos theory

Chaos theory describes the behavior of certain dynamical systems — that is, systems whose state evolves with
time — that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the
butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations
in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these
systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with
no random elements involved. This behavior is known as deterministic chaos, or simply chaos.

Complex systems

Complex systems is a scientific field, which studies the common properties of systems considered complex in
nature, society and science. It is also called complex systems theory, complexity science, study of complex
systems and/or sciences of complexity. The key problems of such systems are difficulties with their formal
modeling and simulation. From such perspective, in different research contexts complex systems are defined
on the base of their different attributes.

The study of complex systems is bringing new vitality to many areas of science where a more typical
reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a
research approach to problems in many diverse disciplines including neurosciences, social sciences,
meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation,
economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves.

Dynamical systems theory 40

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with influencing the
behavior of dynamical systems.

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and
related problems. Its initial development was motivated by problems of statistical physics.

Functional analysis

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of
vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in
particular transformations of functions, such as the Fourier transform, as well as in the study of differential and
integral equations. This usage of the word functional goes back to the calculus of variations, implying a
function whose argument is a function. Its use in general has been attributed to mathematician and physicist
Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.

Graph dynamical systems

The concept of graph dynamical systems (GDS) can be used to capture a wide range of processes taking place
on graphs or networks. A major theme in the mathematical and computational analysis of GDS is to relate
their structural properties (e.g. the network connectivity) and the global dynamics that result.

Projected dynamical systems

Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where
solutions are restricted to a constraint set. The discipline shares connections to and applications with both the
static world of optimization and equilibrium problems and the dynamical world of ordinary differential
equations. A projected dynamical system is given by the flow to the projected differential equation.

Symbolic dynamics

Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space
consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with
the dynamics (evolution) given by the shift operator.

System dynamics

Dynamical systems theory 41

Topological dynamics

Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic
properties of dynamical systems are studied from the viewpoint of general topology.

Applications
In biomechanics

In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for
modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly
intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual)
that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue,

metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through

T31
generic processes of self-organization found in physical and biological systems.

In cognitive science

Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the
neo-Piagetian theories of cognitive development. It is the belief that cognitive development is best represented by
physical theories rather than theories based on syntax and AI. It also believes that differential equations are the most
appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive
trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description
(via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal
pressures. The language of chaos theory is also frequently adopted.

In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase

transition of cognitive development. Self organization (the spontaneous creation of coherent forms) sets in as activity

levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the

process. These links form the structure of a new state of order in the mind through a process called scalloping (the

repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete,

mi
idiosyncratic and unpredictable.

Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred
to as the A-not-B error.

Notes

[1] Jerome R. Busemeyer (2008), "Dynamic Systems" (http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html). To Appear

in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008.
[2] MIT System Dynamics in Education Project (SDEP) (http://sysdyn.clexchange.org)
[3] Paul S Glaziera, Keith Davidsb, Roger M Bartlettc (2003). "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for

Performance-Oriented Sports Biomechanics Research" (http://www.sportsci.org/jour/03/psg.htm). in: Sportscience 7.

Accessdate=2008-05-08.
[4] Lewis, Mark D. (2000-02-25). "The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development" (http://

home.oise.utoronto.ca/~mlewis/Manuscripts/Promise.pdf) (PDF). Child Development 71 (1): 36^-3. doi:10.1111/1467-8624.00116.

PMID 10836556. . Retrieved 2008-04-04.
[5] Smith, Linda B.; Esther Thelen (2003-07-30). "Development as a dynamic system" (http://www.indiana.edu/~cogdev/labwork/

dynamicsystem.pdf) (PDF). TRENDS in Cognitive Sciences 7 (8): 343-8. doi:10.1016/S1364-6613(03)00156-6. . Retrieved 2008-04-04.

Dynamical systems theory 42

Further reading

• Frederick David Abraham (1990), A Visual Introduction to Dynamical Systems Theory for Psychology, 1990.

• Beltrami, E. J. (1987). Mathematics for dynamic modeling. NY: Academic Press

• Otomar Hajek (1968 }, Dynamical Systems in the Plane.

• Luenberger, D. G. (1979). Introduction to dynamic systems. NY: Wiley.

• Anthony N. Michel, Kaining Wang & Bo Hu (2001), Qualitative Theory of Dynamical Systems: The Role of
Stability Preserving Mappings.

• Padulo, L. & Arbib, M A. (1974). System Theory. Philadelphia: Saunders

• Strogatz, S. H. (1994), Nonlinear dynamics and chaos. Reading, MA: Addison Wesley

External links

• Dynamic Systems (http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html) Encyclopedia of
Cognitive Science entry.

• Definition of dynamical system (http://mathworld.wolfram.com/DynamicalSystem.html) in Math World.

• DSWeb (http://www.dynamicalsystems.org/) Dynamical Systems Magazine

Symbolic dynamics

In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a
discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the
system, with the dynamics (evolution) given by the shift operator. Formally, a Markov partition is used to provide a
finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of
symbols result as a trajectory of the system moves from one of the covering sets to another.

History

The idea goes back to Jacques Hadamard's 1898 paper on the geodesies on surfaces of negative curvature. It was
applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done
by Emil Artin in 1924 (for the system now called Artin billiard), P. J. Myrberg, Paul Koebe, Jakob Nielsen, G. A.
Hedlund.

The first formal treatment was developed by Morse and Hedlund in their 1938 paper. George Birkhoff, Norman
Levinson and M. L. Cartwright— J. E. Littlewood have applied similar methods to qualitative analysis of
nonautonomous second order differential equations.

Claude Shannon used symbolic sequences and shifts of finite type in his 1948 paper A mathematical theory of
communication that gave birth to information theory.

The theory was further advanced in the 1960s and 1970s, notably, in the works of Steve Smale and his school, and of
Yakov Sinai and the Soviet school of ergodic theory. A spectacular application of the methods of symbolic dynamics
is Sharkovskii's theorem about periodic orbits of a continuous map of an interval into itself (1964).

Symbolic dynamics 43

Examples

Concepts such as heteroclinic orbits and homoclinic orbits have a particularly simple representation in symbolic
dynamics.

Applications

Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have
found significant applications in data storage and transmission, linear algebra, the motions of the planets and many
other areas. The distinct feature in symbolic dynamics is that time is measured in discrete intervals. So at each time
interval the system is in a particular state. Each state is associated with a symbol and the evolution of the system is
described by an infinite sequence of symbols — represented effectively as strings. If the system states are not
inherently discrete, then the state vector must be discretized, so as to get a coarse-grained description of the system.

Further reading

• Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems . World Scientific.
ISBN 9971-50-682-3.

• Bruce Kitchens, Symbolic dynamics. One-sided, two-sided and countable state Markov shifts. Universitext,
Springer- Verlag, Berlin, 1998. x+252 pp. ISBN 3-540-62738-3 MR1484730

• Douglas Lind and Brian Marcus, An Introduction to Symbolic Dynamics and Coding . Cambridge University
Press, Cambridge, 1995. xvi+495 pp. ISBN 0-521-55124-2 MR1369092

• M. Morse and G. A. Hedlund, Symbolic Dynamics, American Journal of Mathematics, 60 (1938) 815—866

T31

• G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system . Math. Systems Theory,

Vol. 3, No. 4 (1969) 320-3751

References

[1] http://power.itp.ac.cn/~hao/

[2] http://www.math.washington.edu/SymbolicDynamics/

[3] http://www.springerlink.com/content/k629151862130377/

Molecular dynamics 44

Molecular dynamics

Molecular dynamics (MD) is computer simulation of physical movements by atoms and molecules.

Molecular dynamics simulation is frequently used in the study of proteins and biomolecules, as well as in materials
science. It is tempting, though not entirely accurate, to describe the technique as a "virtual microscope" with high
temporal and spatial resolution. Whereas it is possible to take "still snapshots" of crystal structures and probe
features of the motion of molecules through NMR, no current experimental technique allows access to all the time
scales of motion with atomic resolution. Richard Feynman once said that "If we were to name the most powerful
assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms,
and that everything that living things do can be understood in terms of the jigglings and wigglings of atoms."
Molecular dynamics lets scientists peer into the motion of individual atoms in a way which is not possible in
laboratory experiments.

Molecular dynamics is a specialized discipline of molecular modeling and computer simulation based on statistical
mechanics; the main justification of the MD method is that statistical ensemble averages are equal to time averages
of the system, known as the ergodic hypothesis. MD has also been termed "statistical mechanics by numbers" and
"Laplace's vision of Newtonian mechanics" of predicting the future by animating nature's forces and allowing

insight into molecular motion on an atomic scale. However, long MD simulations are mathematically ill-conditioned,
generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and
parameters, but not eliminated entirely. Furthermore, current potential energy functions (also called force-fields) are,
in many cases, not sufficiently accurate to reproduce the dynamics of molecular systems, so the much more
computationally demanding Ab Initio Molecular Dynamics method must be used. Nevertheless, molecular dynamics
techniques allow detailed time and space resolution into representative behavior in phase space for carefully selected
systems.

Before it became possible to simulate molecular dynamics with computers, some undertook the hard work of trying
it with physical models such as macroscopic spheres. The idea was to arrange them to replicate the properties of a
liquid. J.D. Bernal said, in 1962: "... I took a number of rubber balls and stuck them together with rods of a selection
of different lengths ranging from 2.75 to 4 inches. I tried to do this in the first place as casually as possible, working
in my own office, being interrupted every five minutes or so and not remembering what I had done before the
interruption." Fortunately, now computers keep track of bonds during a simulation.

Because molecular systems generally consist of a vast number of particles, it is in general impossible to find the
properties of such complex systems analytically. When the number of particles interacting is higher than two, the
result is chaotic motion (see n-body problem). MD simulation circumvents the analytical intractability by using
numerical methods. It represents an interface between laboratory experiments and theory, and can be understood as a
"virtual experiment". MD probes the relationship between molecular structure, movement and function. Molecular
dynamics is a multidisciplinary method. Its laws and theories stem from mathematics, physics, and chemistry, and it
employs algorithms from computer science and information theory. It was originally conceived within theoretical
physics in the late 1950s and early 1960s , but is applied today mostly in materials science and the modeling of
biomolecules.

Molecular dynamics

Areas of Application

There is a significant difference
between the focus and methods used by
chemists and physicists, and this is
reflected in differences in the jargon
used by the different fields. In
chemistry and biophysics, the
interaction between the particles is
either described by a "force field"
(classical MD), a quantum chemical
model, or a mix between the two. These
terms are not used in physics, where the
interactions are usually described by the
name of the theory or approximation
being used and called the potential
energy, or just the "potential".

Give atoms initial positions K'- - 1 , choose short At
1

Get forces F= - v V(r®) and a = F/m
T

Move atoms: r<<* 1 >

= ffiJ +V (0 £t

+ V 2 aAt 2 + ...

Move time

forward: f =

t + At

Repeat as long as you need

Highly simplified description of the molecular dynamics simulation algorithm. The

simulation proceeds iteratively by alternatively calculating forces and solving the

equations of motion based on the accelerations obtained from the new forces. In practise,

almost all MD codes use much more complicated versions of the algorithm, including

two steps (predictor and corrector) in solving the equations of motion and many

additional steps for e.g. temperature and pressure control, analysis and output.

Beginning in theoretical physics, the method of MD gained popularity in materials science and since the 1970s also
in biochemistry and biophysics. In chemistry, MD serves as an important tool in protein structure determination and
refinement using experimental tools such as X-ray crystallography and NMR. It has also been applied with limited
success as a method of refining protein structure predictions. In physics, MD is used to examine the dynamics of
atomic-level phenomena that cannot be observed directly, such as thin film growth and ion-subplantation. It is also
used to examine the physical properties of nanotechnological devices that have not or cannot yet be created.

In applied mathematics and theoretical physics, molecular dynamics is a part of the research realm of dynamical
systems, ergodic theory and statistical mechanics in general. The concepts of energy conservation and molecular
entropy come from thermodynamics. Some techniques to calculate conformational entropy such as principal
components analysis come from information theory. Mathematical techniques such as the transfer operator become
applicable when MD is seen as a Markov chain. Also, there is a large community of mathematicians working on
volume preserving, symplectic integrators for more computationally efficient MD simulations.

MD can also be seen as a special case of the discrete element method (DEM) in which the particles have spherical
shape (e.g. with the size of their van der Waals radii.) Some authors in the DEM community employ the term MD
rather loosely, even when their simulations do not model actual molecules.

Design Constraints

Design of a molecular dynamics simulation should account for the available computational power. Simulation size
(n=number of particles), timestep and total time duration must be selected so that the calculation can finish within a
reasonable time period. However, the simulations should be long enough to be relevant to the time scales of the
natural processes being studied. To make statistically valid conclusions from the simulations, the time span
simulated should match the kinetics of the natural process. Otherwise, it is analogous to making conclusions about

how a human walks from less than one footstep. Most scientific publications about the dynamics of proteins and

-9 -6

DNA use data from simulations spanning nanoseconds (10 s) to microseconds (10 s). To obtain these simulations,

several CPU-days to CPU-years are needed. Parallel algorithms allow the load to be distributed among CPUs; an

example is the spatial or force decomposition algorithm [6].

During a classical MD simulation, the most CPU intensive task is the evaluation of the potential (force field) as a
function of the particles' internal coordinates. Within that energy evaluation, the most expensive one is the

Molecular dynamics 46

non-bonded or non-covalent part. In Big O notation, common molecular dynamics simulations scale by 0(ri 2 )if all

pair-wise electrostatic and van der Waals interactions must be accounted for explicitly. This computational cost can

be reduced by employing electrostatics methods such as Particle Mesh Ewald ( 0(nlog(n))), P3M or good spherical cutoff

techniques ( Oin))-

Another factor that impacts total CPU time required by a simulation is the size of the integration timestep. This is the

time length between evaluations of the potential. The timestep must be chosen small enough to avoid discretization

errors (i.e. smaller than the fastest vibrational frequency in the system). Typical timesteps for classical MD are in the

order of 1 femtosecond (10" s). This value may be extended by using algorithms such as SHAKE, which fix the

vibrations of the fastest atoms (e.g. hydrogens) into place. Multiple time scale methods have also been developed,

which allow for extended times between updates of slower long-range forces.

For simulating molecules in a solvent, a choice should be made between explicit solvent and implicit solvent.
Explicit solvent particles (such as the TIP3P, SPC/E and SPC-f water models) must be calculated expensively by the
force field, while implicit solvents use a mean-field approach. Using an explicit solvent is computationally
expensive, requiring inclusion of roughly ten times more particles in the simulation. But the granularity and viscosity
of explicit solvent is essential to reproduce certain properties of the solute molecules. This is especially important to
reproduce kinetics.

In all kinds of molecular dynamics simulations, the simulation box size must be large enough to avoid boundary
condition artifacts. Boundary conditions are often treated by choosing fixed values at the edges (which may cause
artifacts), or by employing periodic boundary conditions in which one side of the simulation loops back to the
opposite side, mimicking a bulk phase.

Microcanonical ensemble (NVE)

In the microcanonical, or NVE ensemble, the system is isolated from changes in moles (N), volume (V) and energy
(E). It corresponds to an adiabatic process with no heat exchange. A microcanonical molecular dynamics trajectory
may be seen as an exchange of potential and kinetic energy, with total energy being conserved. For a system of N
particles with coordinates X an d velocities \F, the following pair of first order differential equations may be
written in Newton's notation as

F(X) = -VU(X) = MV(t)

V(t)=X(t).
The potential energy function JJ(X) of the system is a function of the particle coordinates _X" . It is referred to
simply as the "potential" in Physics, or the "force field" in Chemistry. The first equation comes from Newton's laws;
the force JP acting on each particle in the system can be calculated as the negative gradient of U(X) ■
For every timestep, each particle's position X an d velocity \/may be integrated with a symplectic method such as
Verlet. The time evolution of X an d V* 15 called a trajectory. Given the initial positions (e.g. from theoretical
knowledge) and velocities (e.g. randomized Gaussian), we can calculate all future (or past) positions and velocities.

One frequent source of confusion is the meaning of temperature in MD. Commonly we have experience with
macroscopic temperatures, which involve a huge number of particles. But temperature is a statistical quantity. If
there is a large enough number of atoms, statistical temperature can be estimated from the instantaneous
temperature, which is found by equating the kinetic energy of the system to nk T/2 where n is the number of

degrees of freedom of the system.

A temperature-related phenomenon arises due to the small number of atoms that are used in MD simulations. For
example, consider simulating the growth of a copper film starting with a substrate containing 500 atoms and a
deposition energy of 100 eV. In the real world, the 100 eV from the deposited atom would rapidly be transported
through and shared among a large number of atoms ( ^Ql°or more) with no big change in temperature. When there
are only 500 atoms, however, the substrate is almost immediately vaporized by the deposition. Something similar

Molecular dynamics 47

happens in biophysical simulations. The temperature of the system in NVE is naturally raised when macromolecules
such as proteins undergo exothermic conformational changes and binding.

Canonical ensemble (NVT)

In the canonical ensemble, moles (N), volume (V) and temperature (T) are conserved. It is also sometimes called
constant temperature molecular dynamics (CTMD). In NVT, the energy of endothermic and exothermic processes is
exchanged with a thermostat.

A variety of thermostat methods is available to add and remove energy from the boundaries of an MD system in a
more or less realistic way, approximating the canonical ensemble. Popular techniques to control temperature include
velocity rescaling, the Nose-Hoover thermostat, Nose-Hoover chains, the Berendsen thermostat and Langevin
dynamics. Note that the Berendsen thermostat might introduce the flying ice cube effect, which leads to unphysical
translations and rotations of the simulated system.

It is not trivial to obtain a canonical distribution of conformations and velocities using these algorithms. How this
depends on system size, thermostat choice, thermostat parameters, time step and integrator is the subject of many
articles in the field.

Isothermal-Isobaric (NPT) ensemble

In the isothermal-isobaric ensemble, moles (N), pressure (P) and temperature (T) are conserved. In addition to a
thermostat, a barostat is needed. It corresponds most closely to laboratory conditions with a flask open to ambient
temperature and pressure.

In the simulation of biological membranes, isotropic pressure control is not appropriate. For lipid bilayers, pressure
control occurs under constant membrane area (NPAT) or constant surface tension "gamma" (NPyT).

Generalized ensembles

The replica exchange method is a generalized ensemble. It was originally created to deal with the slow dynamics of
disordered spin systems. It is also called parallel tempering. The replica exchange MD (REMD) formulation tries
to overcome the multiple-minima problem by exchanging the temperature of non-interacting replicas of the system
running at several temperatures.

Potentials in MD simulations

A molecular dynamics simulation requires the definition of a potential function, or a description of the terms by
which the particles in the simulation will interact. In chemistry and biology this is usually referred to as a force field.
Potentials may be defined at many levels of physical accuracy; those most commonly used in chemistry are based on
molecular mechanics and embody a classical treatment of particle-particle interactions that can reproduce structural
and conformational changes but usually cannot reproduce chemical reactions.

The reduction from a fully quantum description to a classical potential entails two main approximations. The first
one is the Born-Oppenheimer approximation, which states that the dynamics of electrons is so fast that they can be
considered to react instantaneously to the motion of their nuclei. As a consequence, they may be treated separately.
The second one treats the nuclei, which are much heavier than electrons, as point particles that follow classical
Newtonian dynamics. In classical molecular dynamics the effect of the electrons is approximated as a single
potential energy surface, usually representing the ground state.

When finer levels of detail are required, potentials based on quantum mechanics are used; some techniques attempt
to create hybrid classical/quantum potentials where the bulk of the system is treated classically but a small region is
treated as a quantum system, usually undergoing a chemical transformation.

Molecular dynamics 48

Empirical potentials

Empirical potentials used in chemistry are frequently called force fields, while those used in materials physics are
called just empirical or analytical potentials.

Most force fields in chemistry are empirical and consist of a summation of bonded forces associated with chemical
bonds, bond angles, and bond dihedrals, and non-bonded forces associated with van der Waals forces and
electrostatic charge. Empirical potentials represent quantum-mechanical effects in a limited way through ad-hoc
functional approximations. These potentials contain free parameters such as atomic charge, van der Waals
parameters reflecting estimates of atomic radius, and equilibrium bond length, angle, and dihedral; these are obtained
by fitting against detailed electronic calculations (quantum chemical simulations) or experimental physical properties
such as elastic constants, lattice parameters and spectroscopic measurements.

Because of the non-local nature of non-bonded interactions, they involve at least weak interactions between all
particles in the system. Its calculation is normally the bottleneck in the speed of MD simulations. To lower the
computational cost, force fields employ numerical approximations such as shifted cutoff radii, reaction field
algorithms, particle mesh Ewald summation, or the newer Particle-Particle Particle Mesh (P3M).

Chemistry force fields commonly employ preset bonding arrangements (an exception being ab-initio dynamics), and
thus are unable to model the process of chemical bond breaking and reactions explicitly. On the other hand, many of
the potentials used in physics, such as those based on the bond order formalism can describe several different
coordinations of a system and bond breaking. Examples of such potentials include the Brenner potential for
hydrocarbons and its further developments for the C-Si-H and C-O-H systems. The ReaxFF potential can be
considered a fully reactive hybrid between bond order potentials and chemistry force fields.

Pair potentials vs. many-body potentials

The potential functions representing the non-bonded energy are formulated as a sum over interactions between the
particles of the system. The simplest choice, employed in many popular force fields, is the "pair potential", in which
the total potential energy can be calculated from the sum of energy contributions between pairs of atoms. An
example of such a pair potential is the non-bonded Lennard-Jones potential (also known as the 6-12 potential), used
for calculating van der Waals forces.

"M = *[(■;) -{7.

Another example is the Born (ionic) model of the ionic lattice. The first term in the next equation is Coulomb's law
for a pair of ions, the second term is the short-range repulsion explained by Pauli's exclusion principle and the final
term is the dispersion interaction term. Usually, a simulation only includes the dipolar term, although sometimes the
quadrupolar term is included as well.

In many-body potentials, the potential energy includes the effects of three or more particles interacting with each
other. In simulations with pairwise potentials, global interactions in the system also exist, but they occur only
through pairwise terms. In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms,
as these interactions are calculated explicitly as a combination of higher-order terms. In the statistical view, the
dependency between the variables cannot in general be expressed using only pairwise products of the degrees of

ri3i

freedom. For example, the Tersoff potential, which was originally used to simulate carbon, silicon and
germanium and has since been used for a wide range of other materials, involves a sum over groups of three atoms,
with the angles between the atoms being an important factor in the potential. Other examples are the embedded-atom
method (EAM) and the Tight-Binding Second Moment Approximation (TBSMA) potentials, where the
electron density of states in the region of an atom is calculated from a sum of contributions from surrounding atoms,

Molecular dynamics 49

and the potential energy contribution is then a function of this sum.

Semi-empirical potentials

Semi-empirical potentials make use of the matrix representation from quantum mechanics. However, the values of
the matrix elements are found through empirical formulae that estimate the degree of overlap of specific atomic
orbitals. The matrix is then diagonalized to determine the occupancy of the different atomic orbitals, and empirical
formulae are used once again to determine the energy contributions of the orbitals.

There are a wide variety of semi-empirical potentials, known as tight-binding potentials, which vary according to the
atoms being modeled.

Polarizable potentials

Most classical force fields implicitly include the effect of polarizability, e.g. by scaling up the partial charges
obtained from quantum chemical calculations. These partial charges are stationary with respect to the mass of the
atom. But molecular dynamics simulations can explicitly model polarizability with the introduction of induced
dipoles through different methods, such as Drude particles or fluctuating charges. This allows for a dynamic
redistribution of charge between atoms which responds to the local chemical environment.

For many years, polarizable MD simulations have been touted as the next generation. For homogenous liquids such

as water, increased accuracy has been achieved through the inclusion of polarizability. Some promising results

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have also been achieved for proteins. However, it is still uncertain how to best approximate polarizability in a

simulation.

Ab-initio methods

In classical molecular dynamics, a single potential energy surface (usually the ground state) is represented in the
force field. This is a consequence of the Born-Oppenheimer approximation. In excited states, chemical reactions or a
more accurate representation is needed, electronic behavior can be obtained from first principles by using a quantum
mechanical method, such as Density Functional Theory. This is known as Ab Initio Molecular Dynamics (AIMD).
Due to the cost of treating the electronic degrees of freedom, the computational cost of this simulations is much
higher than classical molecular dynamics. This implies that AIMD is limited to smaller systems and shorter periods
of time.

Ab-initio quantum-mechanical methods may be used to calculate the potential energy of a system on the fly, as
needed for conformations in a trajectory. This calculation is usually made in the close neighborhood of the reaction
coordinate. Although various approximations may be used, these are based on theoretical considerations, not on
empirical fitting. Ab-initio calculations produce a vast amount of information that is not available from empirical
methods, such as density of electronic states or other electronic properties. A significant advantage of using ab-initio
methods is the ability to study reactions that involve breaking or formation of covalent bonds, which correspond to
multiple electronic states.

A popular software for ab-initio molecular dynamics is the Car-Parrinello Molecular Dynamics (CPMD) package
based on the density functional theory.

Hybrid QM/MM

QM (quantum-mechanical) methods are very powerful. However, they are computationally expensive, while the
MM (classical or molecular mechanics) methods are fast but suffer from several limitations (require extensive
parameterization; energy estimates obtained are not very accurate; cannot be used to simulate reactions where
covalent bonds are broken/formed; and are limited in their abilities for providing accurate details regarding the
chemical environment). A new class of method has emerged that combines the good points of QM (accuracy) and

Molecular dynamics 50

MM (speed) calculations. These methods are known as mixed or hybrid quantum-mechanical and molecular
mechanics methods (hybrid QM/MM). The methodology for such techniques was introduced by Warshel and
coworkers. In the recent years have been pioneered by several groups including: Arieh Warshel (University of
Southern California), Weitao Yang (Duke University), Sharon Hammes-Schiffer (The Pennsylvania State
University), Donald Truhlar and Jiali Gao (University of Minnesota) and Kenneth Merz (University of Florida).

The most important advantage of hybrid QM/MM methods is the speed. The cost of doing classical molecular

dynamics (MM) in the most straightforward case scales 0(n ), where n is the number of atoms in the system. This is
mainly due to electrostatic interactions term (every particle interacts with every other particle). However, use of
cutoff radius, periodic pair-list updates and more recently the variations of the particle-mesh Ewald's (PME) method

has reduced this between O(n) to 0(n ). In other words, if a system with twice as many atoms is simulated then it
would take between two to four times as much computing power. On the other hand the simplest ab-initio

calculations typically scale 0(n ) or worse (Restricted Hartree-Fock calculations have been suggested to scale

2 7

~0(n ' )). To overcome the limitation, a small part of the system is treated quantum-mechanically (typically
active-site of an enzyme) and the remaining system is treated classically.

In more sophisticated implementations, QM/MM methods exist to treat both light nuclei susceptible to quantum
effects (such as hydrogens) and electronic states. This allows generation of hydrogen wave-functions (similar to
electronic wave-functions). This methodology has been useful in investigating phenomena such as hydrogen
tunneling. One example where QM/MM methods have provided new discoveries is the calculation of hydride
transfer in the enzyme liver alcohol dehydrogenase. In this case, tunneling is important for the hydrogen, as it
determines the reaction rate.

Coarse-graining and reduced representations

At the other end of the detail scale are coarse-grained and lattice models. Instead of explicitly representing every
atom of the system, one uses "pseudo-atoms" to represent groups of atoms. MD simulations on very large systems
may require such large computer resources that they cannot easily be studied by traditional all-atom methods.
Similarly, simulations of processes on long timescales (beyond about 1 microsecond) are prohibitively expensive,
because they require so many timesteps. In these cases, one can sometimes tackle the problem by using reduced
representations, which are also called coarse-grained models.

Examples for coarse graining (CG) methods are discontinuous molecular dynamics (CG-DMD) and

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Go-models. Coarse-graining is done sometimes taking larger pseudo-atoms. Such united atom approximations

have been used in MD simulations of biological membranes. The aliphatic tails of lipids are represented by a few

pseudo-atoms by gathering 2 to 4 methylene groups into each pseudo-atom.

The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the
model to appropriate experimental data or all-atom simulations. Ideally, these parameters should account for both
enthalpic and entropic contributions to free energy in an implicit way. When coarse-graining is done at higher levels,
the accuracy of the dynamic description may be less reliable. But very coarse-grained models have been used
successfully to examine a wide range of questions in structural biology.

Examples of applications of coarse-graining in biophysics:

• protein folding studies are often carried out using a single (or a few) pseudo-atoms per amino acid;

• DNA supercoiling has been investigated using 1-3 pseudo-atoms per basepair, and at even lower resolution;

• Packaging of double-helical DNA into bacteriophage has been investigated with models where one pseudo-atom
represents one turn (about 10 basepairs) of the double helix;

• RNA structure in the ribosome and other large systems has been modeled with one pseudo-atom per nucleotide.

The simplest form of coarse-graining is the "united atom" (sometimes called "extended atom") and was used in most
early MD simulations of proteins, lipids and nucleic acids. For example, instead of treating all four atoms of a CH
methyl group explicitly (or all three atoms of CH methylene group), one represents the whole group with a single

Molecular dynamics 5 1

pseudo-atom. This pseudo-atom must, of course, be properly parameterized so that its van der Waals interactions
with other groups have the proper distance-dependence. Similar considerations apply to the bonds, angles, and
torsions in which the pseudo-atom participates. In this kind of united atom representation, one typically eliminates
all explicit hydrogen atoms except those that have the capability to participate in hydrogen bonds ("polar
hydrogens"). An example of this is the Charmm 19 force-field.

The polar hydrogens are usually retained in the model, because proper treatment of hydrogen bonds requires a
reasonably accurate description of the directionality and the electrostatic interactions between the donor and acceptor
groups. A hydroxyl group, for example, can be both a hydrogen bond donor and a hydrogen bond acceptor, and it
would be impossible to treat this with a single OH pseudo-atom. Note that about half the atoms in a protein or
nucleic acid are nonpolar hydrogens, so the use of united atoms can provide a substantial savings in computer time.

Examples of applications

Molecular dynamics is used in many fields of science.

• First macromolecular MD simulation published (1977, Size: 500 atoms, Simulation Time: 9.2 ps=0.0092 ns,
Program: CHARMM precursor) Protein: Bovine Pancreatic Trypsine Inhibitor. This is one of the best studied
proteins in terms of folding and kinetics. Its simulation published in Nature magazine paved the way for

["221

understanding protein motion as essential in function and not just accessory.

• MD is the standard method to treat collision cascades in the heat spike regime, i.e. the effects that energetic
neutron and ion irradiation have on solids an solid surfaces.

The following two biophysical examples are not run-of-the-mill MD simulations. They illustrate notable efforts to
produce simulations of a system of very large size (a complete virus) and very long simulation times (500
microseconds):

• MD simulation of the complete satellite tobacco mosaic virus (STMV) (2006, Size: 1 million atoms, Simulation
time: 50 ns, program: NAMD) This virus is a small, icosahedral plant virus which worsens the symptoms of
infection by Tobacco Mosaic Virus (TMV). Molecular dynamics simulations were used to probe the mechanisms
of viral assembly. The entire STMV particle consists of 60 identical copies of a single protein that make up the
viral capsid (coating), and a 1063 nucleotide single stranded RNA genome. One key finding is that the capsid is
very unstable when there is no RNA inside. The simulation would take a single 2006 desktop computer around 35
years to complete. It was thus done in many processors in parallel with continuous communication between
them. [25]

• Folding Simulations of the Villin Headpiece in All- Atom Detail (2006, Size: 20,000 atoms; Simulation time: 500
|as = 500,000 ns, Program: folding® home) This simulation was run in 200,000 CPU's of participating personal
computers around the world. These computers had the folding© home program installed, a large-scale distributed
computing effort coordinated by Vijay Pande at Stanford University. The kinetic properties of the Villin
Headpiece protein were probed by using many independent, short trajectories run by CPU's without continuous
real-time communication. One technique employed was the Pfold value analysis, which measures the probability
of folding before unfolding of a specific starting conformation. Pfold gives information about transition state
structures and an ordering of conformations along the folding pathway. Each trajectory in a Pfold calculation can
be relatively short, but many independent trajectories are needed.

Molecular dynamics 52

Molecular dynamics algorithms
Integrators

• Verlet-Stoermer integration

• Runge-Kutta integration

• Beeman's algorithm

• Gear predictor - corrector

• Constraint algorithms (for constrained systems)

• Symplectic integrator

Short-range interaction algorithms

• Cell lists

• Verlet list

• Bonded interactions

Long-range interaction algorithms

• Ewald summation

• Particle Mesh Ewald (PME)

• Particle-Particle Particle Mesh P3M

• Reaction Field Method

Parallelization strategies

• Domain decomposition method (Distribution of system data for parallel computing)

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• Molecular Dynamics - Parallel Algorithms

Major software for MD simulations

AutoDock suite of automated docking tools,

Autodock Vina improved local search algorithm, suite of automated docking tools,

Abalone (classical, implicit water)

ABINIT (DFT)

ACEMD [28] (running on NVIDIA GPUs: heavily optimized with CUD A)

[291
ADUN (classical, P2P database for simulations)

AMBER (classical)

Ascalaph (classical, GPU accelerated)

CASTEP (DFT)

CPMD (DFT)

CP2K [31] (DFT)

CHARMM (classical, the pioneer in MD simulation, extensive analysis tools)

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COSMOS (classical and hybrid QM/MM, quantum-mechanical atomic charges with BPT)

Desmond (classical, parallelization with up to thousands of CPU's)

Culgi [33] (classical, OPLS-AA, Dreiding, Nerd, and TraPPE-UA force fields)

DL_POLY [34] (classical)

ESPResSo (classical, coarse-grained, parallel, extensible)

Fireball [35] (tight-binding DFT)

GROMACS (classical)

GROMOS (classical)

Molecular dynamics 53

GULP (classical)

Hippo (classical)

HOOMD-Blue [37] (classical, accelerated by NVIDIA GPUs, heavily optimized with CUDA)

HO]

Kalypso MD simulation of atomic collisions in solids

LAMMPS (classical, large-scale with spatial-decomposition of simulation domain for parallelism)

[39]
LPMD Las Palmeras Molecular Dynamics: flexible an modular MD.

MacroModel (classical)

MACSIMUS (classical, polarizability, thread-based parallelization)

MDynaMix (classical, parallel)

MOLDY [41] (classical, parallel) latest release [42]

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Materials Studio (Forcite MD using COMPASS, Dreiding, Universal, cvff and pcff forcefields in serial or

parallel, QMERA (QM+MD), ONESTEP (DFT), etc.)

MOSCITO (classical)

NAMD (classical, parallelization with up to thousands of CPU's)

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nano-Material Simulation Toolkit

T451
NEWTON-X (ab initio, surface-hopping dynamics)

OR AC (classical)

ProtoMol (classical, extensible, includes multigrid electrostatics)

PWscf (DFT)

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RedMD (coarse-grained simulations package on GNU licence)

S/PHI/nX [48] (DFT)

SIESTA (DFT)

Tremolo-X

VASP (DFT)

TINKER (classical)

YASARA [49] (classical)

XMD (classical)

Related software

Avizo - 3d visualization and analysis software.

BOSS - MC in OPLS

Chimera - Molecular visualization and analysis package, including trajectory support.

esra - Lightweight molecular modeling and analysis library (Java/Jython/Mathematica).

Molecular Workbench - Interactive molecular simulations on your desktop.

T521

Packmol Package for building starting configurations for MD in an automated fashion.

[531
Punto is a freely available visualisation tool for particle simulations.

PyMol - Molecular Visualization software written in python.

Sirius - Molecular modeling, analysis and visualization of MD trajectories.

VMD - MD simulation trajectories can be visualized and analyzed.

Molecular dynamics 54

Specialized hardware for MD simulations

• Anton - A specialized, massively parallel supercomputer designed to execute MD simulations.

• MDGRAPE - A special purpose system built for molecular dynamics simulations, especially protein structure
prediction.

References

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Molecular dynamics

[25

[26

Freddolino P, Arkhipov A, Larson SB, McPherson A, Schulten K. "Molecular dynamics simulation of the Satellite Tobacco Mosaic Virus
(STMV)" (http://www.ks.uiuc.edu/Research/STMV/). Theoretical and Computational Biophysics Group. University of Illinois at Urbana
Champaign. .

The Folding@Home Project (http://folding.stanford.edu/) and recent papers (http://folding.stanford.edu/papers.html) published using
trajectories from it. Vijay Pande Group. Stanford University

7www.cs.sandia.gov/~sjplimp/md.html

'/www. multiscalelab. org/acemd

Vcbbl. imim. es/ Adun

7www.biomolecular-modeling.com/Products.html

7cp2k.berlios.de/

7www.cosmos-software.de/ce_intro.html

'/www. culgi. com/

'/www. ccp5 .ac. uk/DL_POL Y/

7fireball-dft.org

7www.biowerkzeug.com/

7codeblue.umich.edu/hoomd-blue/

7sites.google.com/site/KalypsoSimulation/

7www.gnm.cl/lpmd

7www.vscht.cz/fch/software/macsimus/index.html

'/www. ccp5 .ac. uk/ moldy/moldy, html

7ccpforge.cse.rl.ac.uk/gf/project/moldy/frs

7accelrys.com/products/materials-studio/

7nanohub.org/resources/matsimtk

7www.univie.ac.at/newtonx/

'/protomol. sourceforge.net/

7bionano.icm.edu.pl/software/redmd

'/www. sphinxlib.de

7www.yasara.org

7esra.sourceforge.net/cgi-bin/index.cgi

7mw.concord.org/modeler/

7www.ime.unicamp.br/~martinez/packmol

7sourceforge.net/projects/punto/

[27]

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General references

• M. P. Allen, D. J. Tildesley (1989) Computer simulation of liquids. Oxford University Press. ISBN
0-19-855645-4.

• J. A. McCammon, S. C. Harvey (1987) Dynamics of Proteins and Nucleic Acids. Cambridge University Press.
ISBN 0521307503 (hardback).

• D. C. Rapaport (1996) The Art of Molecular Dynamics Simulation. ISBN 0-521-44561-2.

• Frenkel, Daan; Smit, Berend (2002) [2001]. Understanding Molecular Simulation : from algorithms to
applications. San Diego, California: Academic Press. ISBN 0-12-267351-4.

• J. M. Haile (2001) Molecular Dynamics Simulation: Elementary Methods. ISBN 0-471-18439-X

• R. J. Sadus, Molecular Simulation of Fluids: Theory, Algorithms and Object-Orientation, 2002, ISBN
0-444-51082-6

• Oren M. Becker, Alexander D. Mackerell Jr, Benoit Roux, Masakatsu Watanabe (2001) Computational
Biochemistry and Biophysics. Marcel Dekker. ISBN 0-8247-0455-X.

• Andrew Leach (2001) Molecular Modelling: Principles and Applications. (2nd Edition) Prentice Hall. ISBN
978-0582382107.

• Tamar Schlick (2002) Molecular Modeling and Simulation. Springer. ISBN 0-387-95404-X.

• William Graham Hoover (1991) Computational Statistical Mechanics, Elsevier, ISBN 0-444-88192-1.

• DJ. Evans and G.P. Morriss (2008) Statistical Mechanics of Nonequilibrium Liquids, Second Edition, Cambridge
University Press,ISBN 978-0-521-85791-8.

Molecular dynamics

External links

• The Blue Gene Project (http://researchweb.watson.ibm.com/bluegene/) (IBM)JawBreakers.org

• D. E. Shaw Research (http://deshawresearch.com/) (D. E. Shaw Research)

• Molecular Physics (http://www.tandf.co.uk/journals/titles/00268976.asp)

• Statistical mechanics of Nonequilibrium Liquids (http://www.phys.unsw.edu.au/~gary/book.html) Lecture
Notes on non-equilibrium MD

• Introductory Lecture on Classical Molecular Dynamics (http://www.fz-juelich.de/nic-series/volumelO/
sutmann.pdf) by Dr. Godehard Sutmann, NIC, Forschungszentrum Julich, Germany

• Introductory Lecture on Ab Initio Molecular Dynamics and Ab Initio Path Integrals (http://www.fz-juelich.de/
nic-series/volumel0/tuckerman2.pdf) by Mark E. Tuckerman, New York University, USA

• Introductory Lecture on Ab initio molecular dynamics: Theory and Implementation (http://www.fz-juelich.de/
nic-series/Volumel/marx.pdf) by Dominik Marx, Ruhr-Universitat Bochum and Jiirg Hutter, Universitat Zurich

• Online course on (MSE 597G) An Introduction to Molecular Dynamics (http://nanohub.org/resources/5838)
by Alejandro Strachan

• Lecture Notes on Short Course on Molecular Dynamics Simulation Ashlie Martini (2009) (http://nanohub.org/
resources/7570)

Molecular modeling on GPU

Molecular modeling on GPU is the technique of using a graphics
processing unit (GPU) for molecular simulations.

In 2007, NVIDIA introduced video cards that could be used not only to
show graphics but also for scientific calculations. These cards include
many arithmetic units (currently up to 512) working in parallel. Long
before this event, the computational power of video cards was used to
accelerate calculations. What was new is that stream processing made
it possible to develop parallel programs in a high-level language. This
technology substantially simplified programming by enabling
programs to be written in C/C++.

[2] [3] [4] [5] [6]

Quantum chemistry calculations

and molecular

Ionic liquid simulation on GPU (Ascalaph
Designer)

mechanics simulations (molecular modeling in terms of

classical mechanics) are among beneficial applications of this technology. The video cards can accelerate the
calculations tens of times. Thus, a PC with such a card has the power similar to that of a cluster of workstations
based on the common processors.

Molecular modeling on GPU 57

References

[1] John E. Stone, James C. Phillips, Peter L. Freddolino, David J. Hardy 1, Leonardo G. Trabuco, Klaus Schulten (2007). "Accelerating

molecular modeling applications with graphics processors". Journal of Computational Chemistry 28 (16): 2618—2640. doi: 10. 1002/jcc. 20829.

PMID 17894371.
[2] Koji Yasuda (2008). "Accelerating Density Functional Calculations with Graphics Processing Unit". J. Chem. Theory Comput. 4 (8):

1230-1236. doi:10.1021/ct8001046.
[3] Koji Yasuda (2008). "Two-electron integral evaluation on the graphics processor unit". Journal of Computational Chemistry: 29 (3): 334—342.

doi:10.1002/jcc.20779. PMID 17614340.
[4] Leslie Vogt, Roberto Olivares-Amaya, Sean Kermes, Yihan Shao, Carlos Amador-Bedolla and Alan Aspuru-Guzik (2008). "Accelerating

Resolution-of-the-Identity Second-Order M0ller-Plesset Quantum Chemistry Calculations with Graphical Processing Units". / Phys. Chem.

A 112 (10): 2049-2057. doi:10.1021/jp0776762. PMID 18229900.
[5] Ivan S. Ufimtsev and Todd J. Martinez (2008). "Quantum Chemistry on Graphical Processing Units. 1. Strategies for Two-Electron Integral

Evaluation". J. Chem. Theo. Comp. 4 (2): 222-231. doi:10.1021/ct700268q.
[6] Ivan S. Ufimtsev and Todd J. Martinez (2008). "Graphical Processing Units for Quantum Chemistry". Comp. Sci. Eng. 10 (6): 26—34.

doi: 10. 1 109/MCSE.2008. 148.
[7] Joshua A. Anderson, Chris D. Lorenz, A. Travesset (2008). "General Purpose Molecular Dynamics Simulations Fully Implemented on

Graphics Processing Units". Journal of Computational Physics 227 (10): 5342-5359. doi:10.1016/j.jcp.2008.01.047.
[8] Christopher I. Rodrigues, David J. Hardy, John E. Stone, Klaus Schulten, and Wen-Mei W. Hwu. (2008). "GPU acceleration of cutoff pair

potentials for molecular modeling applications.". In CF'08: Proceedings of the 2008 conference on Computing frontiers, New York, NY, USA:

273-282.
[9] Peter H. Colberg, Felix Hofling (2009). "Accelerating glassy dynamics using graphics processing units.".

arXiv:0912.3824 [physics. comp-ph].

External links

• More links for MD on GPUs (http://www.nvidia.com/object/molecular_dynamics.html)

GPU accelerated software

Programs

Abalone

AceMD (http://www. acellera.com/index. php?arg=acemd) the biomolecular MD package used by GPUGRID

AMBER on GPUs version (http://ambermd.org/gpus/)

Ascalaph (http://www.biomolecular-modeling.com/Products.html) on GPUs version — Ascalaph Liquid GPU

(http://www.biomolecular-modeling.com/Ascalaph/Ascalaph-Liquid.html)

BigDFT Ab initio program based on wavelet

GROMACS on GPUs version (https://simtk.org/project/xml/downloads. xml?group_id=161#package_id600)

HALMD (http://colberg.org/research/halmd/) — Highly Accelerated Large-scale MD package

HOOMD (http://codeblue.umich.edu/hoomd-blue/index.html) — Highly Optimized Object Oriented

Molecular Dynamics

LAMMPS on GPUs version — gpulammps (http://code.google.eom/p/gpulammps/)

TeraChem - Quantum chemistry and ab initio Molecular Dynamics

VMD & NAMD on GPUs versions (http://www.ks.uiuc.edu/Research/gpu/)

Molecular modeling on GPU 58

API

• OpenMM (http://simtk.org/home/openmm/) — an API for accelerating molecular dynamics on GPUs, vl.O
provides GPU-accelerated version of GROMACS

Distributed computing projects

• GPUGRID (http://www.gpugrid.net/) distributed supercomputing infrastructure

• Folding@Home (http://folding.stanford.edu/) distributed computing project

Monte Carlo Methods

Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated
random sampling to compute their results. Monte Carlo methods are often used in simulating physical and
mathematical systems. These methods are most suited to calculation by a computer and tend to be used when it is
infeasible to compute an exact result with a deterministic algorithm. This method is also used to complement the
theoretical derivations.

Monte Carlo methods are especially useful for simulating systems with many coupled degrees of freedom, such as
fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model). They are used
to model phenomena with significant uncertainty in inputs, such as the calculation of risk in business. They are
widely used in mathematics, for example to evaluate multidimensional definite integrals with complicated boundary
conditions. When Monte Carlo simulations have been applied in space exploration and oil exploration, their
predictions of failures, cost overruns and schedule overruns are routinely better than human intuition or alternative
"soft" methods. [2]

The Monte Carlo method was coined in the 1940s by John von Neumann and Stanislaw Ulam, while they were
working on nuclear weapon projects in the Los Alamos National Laboratory. It was named in homage to Monte
Carlo casino, a famous casino, where Ulam's uncle would often gamble away his money.

Introduction

Monte Carlo methods vary, but tend to follow a particular pattern:

1 . Define a domain of possible inputs.

2. Generate inputs randomly from a probability distribution over the domain.

3. Perform a deterministic computation on the inputs.

4. Aggregate the results.

For example, given that a circle inscribed in a square and the square itself have a ratio of areas that is jt/4, the value
of jt can be approximated using a Monte Carlo method:

1 . Draw a square on the ground, then inscribe a circle within it.

2. Uniformly scatter some objects of uniform size (grains of rice or sand) over the square.

3. Count the number of objects inside the circle and the total number of objects.

4. The ratio of the two counts is an estimate of the ratio of the two areas, which is jt/4. Multiply the result by 4 to
estimate jt.

In this procedure the domain of inputs is the square that circumscribes our circle. We generate random inputs by
scattering grains over the square then perform a computation on each input (test whether it falls within the circle).
Finally, we aggregate the results to obtain our final result, the approximation of jt.

To get an accurate approximation for jt this procedure should have two other common properties of Monte Carlo
methods. First, the inputs should truly be random. If grains are purposefully dropped into only the center of the

Monte Carlo Methods 59

circle, they will not be uniformly distributed, and so our approximation will be poor. Second, there should be a large
number of inputs. The approximation will generally be poor if only a few grains are randomly dropped into the
whole square. On average, the approximation improves as more grains are dropped.

History

Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem
and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlo simulations invert this
approach, solving deterministic problems using a probabilistic analog (see Simulated annealing).

An early variant of the Monte Carlo method can be seen in the Buffon's needle experiment, in which n can be
estimated by dropping needles on a floor made of parallel strips of wood. In the 1930s, Enrico Fermi first
experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it.

In 1946, physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that
neutrons would likely travel through various materials. Despite having most of the necessary data, such as the
average distance a neutron would travel in a substance before it collided with an atomic nucleus or how much energy
the neutron was likely to give off following a collision, the problem could not be solved with analytical calculations.
Stanislaw Ulam had the idea of using random experiments. He recounts his inspiration as follows:

The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question
which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was
what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending
a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical
method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count
the number of successful plays. This was already possible to envisage with the beginning of the new era of fast
computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical
physics, and more generally how to change processes described by certain differential equations into an
equivalent form interpretable as a succession of random operations. Later [in 1946, I] described the idea to
John von Neumann, and we began to plan actual calculations.

—Stanislaw Ulam

Being secret, the work of von Neumann and Ulam required a code name. Von Neumann chose the name "Monte
Carlo". The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to
gamble. Using lists of "truly" random numbers was extremely slow, von Neumann developed a form of

making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude,
von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted
that when it went awry it did so obviously, unlike methods which could be subtly incorrect.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by
the computational tools at the time. In the 1950s they were used at Los Alamos for early work relating to the
development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and
operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible
for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide
application in many different fields.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the
development of pseudorandom number generators, which were far quicker to use than the tables of random numbers
that had been previously used for statistical sampling.

Monte Carlo Methods 60

Definitions

ro]

There is no consensus on how Monte Carlo should be defined. For example, Ripley defines most probabilistic
modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo
statistical tests. Sawilowsky distinguishes between a simulation, Monte Carlo method, and a Monte Carlo
simulation. A simulation is a fictitious representation of reality. A Monte Carlo method is a technique that can be
used to solve a mathematical or statistical problem. A Monte Carlo simulation uses repeated sampling to determine
the properties of some phenomenon. Examples:

• Drawing a pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin:
If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50
designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.

• The area of an irregular figure inscribed in a unit square can be determined by throwing darts at the square and
computing the ratio of hits within the irregular figure to the total number of darts thrown. This is a Monte Carlo
method of determining area, but not a simulation.

• Drawing a large number of pseudo-random uniform variables from the interval [0,1], and assigning values less
than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of
repeatedly tossing a coin.

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of
radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be
described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same
computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural
sampling."

Monte Carlo and random numbers

Interestingly, Monte Carlo simulation methods do not always require truly random numbers to be useful — while for
some applications, such as primality testing, unpredictability is vital. Many of the most useful techniques use
deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually
necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain
sense.

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that
the numbers are uniformly distributed or follow another desired distribution when a large enough number of
elements of the sequence are considered is one of the simplest, and most common ones.

Sawilowsky lists the characteristics of a high quality Monte Carlo simulation:

• the (pseudo-random) number generator has certain characteristics (e. g., a long "period" before the sequence
repeats)

the (pseudo-random) number generator produces values that pass tests for randomness
there are enough samples to ensure accurate results
the proper sampling technique is used
the algorithm used is valid for what is being modeled
it simulates the phenomenon in question.

Monte Carlo Methods 61

Monte Carlo simulation versus "what if" scenarios

There are ways of using probabilities that are definitely not Monte Carlo simulations — for example, deterministic
modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate.
Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.

By contrast, Monte Carlo simulations sample probability distribution for each variable to produce hundreds or

[121

thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring. For
example, a comparison of a spreadsheet cost construction model run using traditional "what if scenarios, and then
run again with Monte Carlo simulation and Triangular probability distributions shows that the Monte Carlo analysis
has a narrower range than the "what if analysis. This is because the "what if analysis gives equal weight to all
scenarios (see quantifying uncertainty in corporate finance).

Applications

Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and
systems with a large number of coupled degrees of freedom. Areas of application include:

Physical sciences

Monte Carlo methods are very important in computational physics, physical chemistry, and related applied fields,
and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and
aerodynamic forms. In statistical physics Monte Carlo molecular modeling is an alternative to computational

molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and

ri3i
polymer systems. Quantum Monte Carlo methods solve the many-body problem for quantum systems. In

experimental particle physics, Monte Carlo methods are used for designing detectors, understanding their behavior

and comparing experimental data to theory. In astrophysics, they are used to model the evolution of galaxies.

Monte Carlo methods are also used in the ensemble models that form the basis of modern weather forecasting.

Engineering

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in
process design. The need arises from the interactive, co-linear and non-linear behavior of typical process
simulations. For example,

• in microelectronics engineering, Monte Carlo methods are applied to analyze correlated and uncorrected
variations in analog and digital integrated circuits. This enables designers to estimate realistic 3— sigma corners
and effectively optimize circuit yields.

• in geostatistics and geometallurgy, Monte Carlo methods underpin the design of mineral processing flowsheets
and contribute to quantitative risk analysis.

• impacts of pollution are simulated and diesel compared with petrol.

• In autonomous robotics, Monte Carlo localization can be used to determine the position of a robot, it is often
applied to stochastic filters such as the Kalman filter or Particle filter which form the heart of the SLAM (
simultaneous Localisation and Mapping ) algorithm.

Monte Carlo Methods 62

Computational Biology

Monte Carlo methods are used in computational biology, such for as Bayesian inference in phylogeny.

Biological systems such as proteins membranes, images of cancer, are being studied by means of computer
simulations.

The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy.
Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical
reaction is happening for instance. We can also conduct thought experiments when the physical experiments are not
feasible, for instance breaking bonds, introducing impurities at specific sites, changing the local/global structure, or
introducing external fields.

Applied statistics

In applied statistics, Monte Carlo methods are generally used for two purposes:

1. To compare competing statistics for small samples under realistic data conditions. Although Type I error and
power properties of statistics can be calculated for data drawn from classical theoretical distributions {e.g., normal
curve, Cauchy distribution) for asymptotic conditions (/. e, infinite sample size and infinitesimally small treatment
effect), real data often do not have such distributions.

2. To provide implementations of hypothesis tests that are more efficient than exact tests such as permutation tests
(which are often impossible to compute) while being more accurate than critical values for asymptotic
distributions.

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An
approximate randomization test is based on a specified subset of all permutations (which entails potentially
enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a
specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn
twice — or more frequently — for the efficiency of not having to track which permutations have already been
selected).

Monte Carlo Methods

Games

■""■*

° 1

o
o

Random shots

Algorithms

Outcome

Monte Carlo tree search applied to a game of

Battleship. Initially the algorithm takes random

shots, but as possible states are eliminated, the

shots can be more selective. As a crude example,

if a ship is hit (figure A), then adjacent squares

become much higher priorities (figures B and C).

Monte Carlo methods have recently been incorporated in algorithms
for playing games that have outperformed previous algorithms in
games like Go and Battleship. These algorithms employ Monte Carlo
tree search. Possible algorithms are organized in a tree and a large
number of random simulations are used to estimate the long-term
potential of each move. A black box simulator represents the
opponent's moves. In games like Battleship, where there is only limited
knowledge of the state of the system (i.e., the positions of the ships), a
belief state is constructed consisting of probabilities for each state and
then initial states are sampled for running simulations. The belief state
is updated as the game proceeds, as in the figure. On a 10 x 10 grid, in
which the total possible number of moves is 100, one algorithm sank
all the ships 50 moves faster, on average, than random play

[21]

One of the main problems that this approach has in game playing is
that it sometimes misses an isolated, very good move. These
approaches are often strong strategically but weak tactically, as tactical
decisions tend to rely on a small number of crucial moves which are
easily missed by the randomly searching Monte Carlo algorithm.

Design and visuals

Monte Carlo methods have also proven efficient in solving coupled
integral differential equations of radiation fields and energy transport,
and thus these methods have been used in global illumination
computations which produce photo-realistic images of virtual 3D models, with applications in video games,

architecture, design, computer generated films, and cinematic special effects

[22]

Finance and business

Monte Carlo methods in finance are often used to calculate the value of companies, to evaluate investments in
projects at a business unit or corporate level, or to evaluate financial derivatives. They can be used to model project
schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to
determine outcomes for the overall project.

Telecommunications

When planning a wireless network, design must be proved to work for a wide variety of scenarios that depend
mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically
used to generate these users and their states. The network performance is then evaluated and, if results are not
satisfactory, the network design goes through an optimization process.

Monte Carlo Methods

Use in mathematics

In general, Monte Carlo methods are used in mathematics to solve various problems by generating suitable random
numbers and observing that fraction of the numbers which obeys some property or properties. The method is useful
for obtaining numerical solutions to problems which are too complicated to solve analytically. The most common
application of the Monte Carlo method is Monte Carlo integration.

Integration

Deterministic numerical integration algorithms work well in a small
number of dimensions, but encounter two problems when the functions
have many variables. First, the number of function evaluations needed
increase rapidly with the number of dimensions. For example, if 10
evaluations provide adequate accuracy in one dimension, then 10
points are needed for 100 dimensions — far too many to be computed.
This is called the curse of dimensionality. Second, the boundary of a
multidimensional region may be very complicated, so it may not be
feasible to reduce the problem to a series of nested one-dimensional
integrals. 100 dimensions is by no means unusual, since in many
physical problems, a "dimension" is equivalent to a degree of freedom.

samples area error (%) CLir - ve p^
10 0.2000 -6000 random IT^T

0.9-

0.8-

0.7-

0.6-

0.5-

y •

0.4-

y •

0.3-

0.2-

o.i-

0,1 0.2 0,3 0.4 0,5 0,6 0.7 0.8 0.9 1

Monte-Carlo integration works by comparing

random points with the value of the function

10 -

— ■— real error
predicted

error

10 100

1,000 10,000 100,000 1,000,000

samples

Errors reduce

by a factor of \j\f~N

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in
question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and
taking some kind of average of the function values at these points. By the law of large numbers, this method will
display l/v^convergence — i.e., quadrupling the number of sampled points will halve the error, regardless of the
number of dimensions .

A refinement of this method, known as importance sampling in statistics, involves sampling the points randomly, but
more frequently where the integrand is large. To do this precisely one would have to already know the integral, but
one can approximate the integral by an integral of a similar function or use adaptive routines such as Stratified

[241 [25]

sampling, recursive stratified sampling, adaptive umbrella sampling or the VEGAS algorithm.

A similar approach, the quasi-Monte Carlo method, uses low-discrepancy sequences. These sequences "fill" the area
better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on
the integral more quickly.

Monte Carlo Methods 65

Another class of methods for sampling points in a volume is to simulate random walks over it (Markov chain Monte
Carlo). Such methods include the Metropolis-Hastings algorithm, Gibbs sampling and the Wang and Landau
algorithm.

Optimization

Another powerful and very popular application for random numbers in numerical simulation is in numerical
optimization. The problem is to minimize (or maximize) functions of some vector that often has a large number of
dimensions. Many problems can be phrased in this way: for example, a computer chess program could be seen as
trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the traveling
salesman problem the goal is to minimize distance traveled. There are also applications to engineering design, such
as multidisciplinary design optimization.

Most Monte Carlo optimization methods are based on random walks. Essentially, the program moves randomly on a
multi-dimensional surface, preferring moves that reduce the function, but sometimes moving "uphill".

Inverse problems

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space.
This probability distribution combines a priori information with new information obtained by measuring some
observable parameters (data). As, in the general case, the theory linking data with model parameters is nonlinear, the
a posteriori probability in the model space may not be easy to describe (it may be multimodal, some moments may
not be defined, etc.).

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we
normally also wish to have information on the resolution power of the data. In the general case we may have a large
number of model parameters, and an inspection of the marginal probability densities of interest may be impractical,
or even useless. But it is possible to pseudorandomly generate a large collection of models according to the posterior
probability distribution and to analyze and display the models in such a way that information on the relative
likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient
Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available.

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method
that allows analysis of (possibly highly nonlinear) inverse problems with complex a priori information and data with
an arbitrary noise distribution.

Computational mathematics

Monte Carlo methods are useful in many areas of computational mathematics, where a "lucky choice" can find the
correct result. A classic example is Rabin's algorithm for primality testing: for any n which is not prime, a random x
has at least a 75% chance of proving that n is not prime. Hence, if n is not prime, but x says that it might be, we have
observed at most a l-in-4 event. If 10 different random x say that "« is probably prime" when it is not, we have
observed a one-in-a-million event. In general a Monte Carlo algorithm of this kind produces one correct answer with
a guarantee n is composite, and x proves it so, but another one without, but with a guarantee of not getting this
answer when it is wrong too often — in this case at most 25% of the time. See also Las Vegas algorithm for a related,
but different, idea.

Monte Carlo Methods

Notes

[1] Hubbart2007

[2] Hubbard 2009

[3] Metropolis 1987

[4] Kalos & Whitlock 2008

[5] Eckardt 1987

[6] Grinstead & Snell 1997

[7] Anderson 1986

[8] Ripley 1987

[9] Sawilowsky 2003

[10

[II
[12
[13
[14
[15
[16
[17
[18
[19
[20
[21
[22
[23
[24

[25

[26

[27

Davenport 1992

Vose 2000, p. 13

Vose 2000, p. 16

Baeurle 2009

MacGillivray & Dodd 1982

Int Panis et al. 2001

Int Panis et al. 2002

Ojeda & et al. 2009,

Milik & Skolnick 1993

Forastero et al. 2010

Sawilowsky & Fahoome 2003

Silver & Veness 2010

Szirmay-Kalos 2008

Press etal. 1996

MEZEI, M (31 December 1986). "Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias". Journal of
Computational Physics 68 (1): 237-248. Bibcode 1987JCoPh..68..237M. doi:doi:10.1016/0021-9991(87)90054-4.

Bartels, Christian; Karplus, Martin (31 December 1997). "Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of
the Potential Energy". The Journal of Physical Chemistry B 102 (5): 865-880. doi: 10.1021/jp972280j.

Mosegaard & Tarantola 1995

Tarantola 2005

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• Berg, Bernd A. (2004). Markov Chain Monte Carlo Simulations and Their Statistical Analysis (With Web-Based
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• Binder, Kurt (1995). The Monte Carlo Method in Condensed Matter Physics. New York: Springer.
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• Caflisch, R. E. (1998). Monte Carlo and quasi-Monte Carlo methods. Acta Numerica. 7. Cambridge University
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• Grinstead, Charles; Snell, J. Laurie (1997). Introduction to Probability. American Mathematical Society.
pp. 10-11.

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• Hubbard, Douglas (2007). How to Measure Anything: Finding the Value of Intangibles in Business. John Wiley &
Sons. p. 46.

• Hubbard, Douglas (2009). The Failure of Risk Management: Why It's Broken and How to Fix It. John Wiley &
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• Kahneman, D.; Tversky, A. (1982). Judgement under Uncertainty: Heuristics and Biases. Cambridge University
Press.

• Kalos, Malvin H; Whitlock, Paula A. (2008). Monte Carlo Methods. Wiley-VCH. ISBN 978-3527407606.

• Kroese, D. P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods (http://www.
montecarlohandbook.org). New York: John Wiley & Sons. pp. 772. ISBN 0470177934.

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• MacKeown, P. Kevin (1997). Stochastic Simulation in Physics. New York: Springer. ISBN 9813083263.

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Geophys. Res. 100 (B7): 12431-12447. Bibcode 1995JGR...10012431M. doi:10.1029/94JB03097.

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Monte Carlo Methods 68

Rubinstein, R. Y.; Kroese, D. P. (2007). Simulation and the Mont Carlo Method (2nd ed.). New York: John Wiley

& Sons. ISBN 9780470177938.

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External links

Overview and reference list (http://mathworld.wolfram.com/MonteCarloMethod.html), Mathworld

Introduction to Monte Carlo Methods (http://www.phy.ornl.gov/csep/CSEP/MC/MC.html), Computational

Science Education Project

The Basics of Monte Carlo Simulations (http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html),

University of Nebraska-Lincoln

Introduction to Monte Carlo simulation (http://office.microsoft.com/en-us/excel-help/

introduction-to-monte-carlo-simulation-HA010282777.aspx) (for Microsoft Excel), Wayne L. Winston

Monte Carlo Methods — Overview and Concept (http://www.brighton-webs.co.uk/montecarlo/concept.asp),

brighton-webs.co.uk

Molecular Monte Carlo Intro (http://www.cooper.edu/engineering/chemechem/monte.html), Cooper Union

Monte Carlo techniques applied in physics (http://www.princeton.edu/~achremos/Appletl-page.htm)

Monte Carlo Method Example (http://waqqasfarooq.com/waqqasfarooq/index. php?option=com_content&

view=article&id=47:monte-carlo&catid=34:statistics&Itemid=53), A step-by-step guide to creating a monte

carlo excel spreadsheet

Pricing using Monte Carlo simulation (http://knol.google.eom/k/giancarlo-vercellino/

pricing-using-monte-carlo-simulation/lld5i2rgd9gn5/3#), a practical example, Prof. Giancarlo Vercellino

Approximate And Double Check Probability Problems Using Monte Carlo method (http://orcik.net/

programming/approximate-and-double-check-probability-problems-using-monte-carlo-method/) at Orcik Dot

Net

Quantum Dynamics

Quantum mechanics, also known as
quantum physics or quantum theory, is a

branch of physics providing a mathematical
description of the dual particle-like and
wave-like behavior and interaction of matter
and energy. Quantum mechanics describes
the time evolution of physical systems via a
mathematical structure called the wave
function. The wave function encapsulates the
probability that the system is to be found in a
given state at a given time. Quantum
mechanics also allows one to calculate the
effect on the system of making measurements
of properties of the system by defining the
effect of those measurements on the wave
function. This leads to the well-known
uncertainty principle as well as enduring
debate over the role of the experimenter,
epitomised in the Schrodinger's Cat thought
experiment.

Quantum mechanics differs significantly
from classical mechanics in its predictions
when the scale of observations becomes
comparable to the atomic and sub-atomic
scale, the so-called quantum realm. However,
many macroscopic properties of systems can
only be fully understood and explained with
the use of quantum mechanics. Phenomena
such as superconductivity, the properties of
materials such as semiconductors and nuclear
and chemical reaction mechanisms observed
mechanics.

v */

:-:

Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical

mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the

position of the ball is represented by a wave (called the wavefunction), with real

part shown in blue and imaginary part in red. Some of the trajectories, such as

C,D,E,F, are standing waves (or "stationary states"). Each standing-wave

frequency is proportional to a possible energy level of the oscillator. This "energy

quantization" does not occur in classical physics, where the oscillator can have

any energy.

as macroscopic behaviour, cannot be explained using classical

The term was coined by Max Planck, and derives from the observation that some physical quantities can be changed
only by discrete amounts, or quanta, as multiples of the Planck constant, rather than being capable of varying
continuously or by any arbitrary amount. For example, the angular momentum, or more generally the action, of an
electron bound into an atom or molecule is quantized. Although an unbound electron does not exhibit quantized
energy levels, one which is bound in an atomic orbital has quantized values of angular momentum. In the context of
quantum mechanics, the wave— particle duality of energy and matter and the uncertainty principle provide a unified
view of the behavior of photons, electrons and other atomic-scale objects.

The mathematical formulations of quantum mechanics are abstract. Similarly, the implications are often
counter-intuitive in terms of classical physics. The centerpiece of the mathematical formulation is the wavefunction

(defined by Schrodinger's wave equation), which describes the probability amplitude of the position and momentum
of a particle. Mathematical manipulations of the wavefunction usually involve the bra-ket notation, which requires

Quantum Dynamics 70

an understanding of complex numbers and linear functionals. The wavefunction treats the object as a quantum
harmonic oscillator and the mathematics is akin to that of acoustic resonance.

Many of the results of quantum mechanics do not have models that are easily visualized in terms of classical
mechanics; for instance, the ground state in the quantum mechanical model is a non-zero energy state that is the
lowest permitted energy state of a system, rather than a traditional classical system that is thought of as simply being
at rest with zero kinetic energy.

Fundamentally, it attempts to explain the peculiar behaviour of matter and energy at the subatomic level — an attempt
which has produced more accurate results than classical physics in predicting how individual particles behave. But
many unexplained anomalies remain.

Historically, the earliest versions of quantum mechanics were formulated in the first decade of the 20th Century,
around the time that atomic theory and the corpuscular theory of light as interpreted by Einstein first came to be
widely accepted as scientific fact; these later theories can be viewed as quantum theories of matter and
electromagnetic radiation.

Following Schrodinger's breakthrough in deriving his wave equation in the mid-1920s, quantum theory was
significantly reformulated away from the old quantum theory, towards the quantum mechanics of Werner
Heisenberg, Max Born, Wolfgang Pauli and their associates, becoming a science of probabilities based upon the
Copenhagen interpretation of Niels Bohr. By 1930, the reformulated theory had been further unified and formalized
by the work of Paul Dirac and John von Neumann, with a greater emphasis placed on measurement, the statistical
nature of our knowledge of reality, and philosophical speculations about the role of the observer.

The Copenhagen interpretation quickly became (and remains) the orthodox interpretation. However, due to the
absence of conclusive experimental evidence there are also many competing interpretations.

Quantum mechanics has since branched out into almost every aspect of physics, and into other disciplines such as
quantum chemistry, quantum electronics, quantum optics and quantum information science. Much 19th Century
physics has been re-evaluated as the classical limit of quantum mechanics and its more advanced developments in
terms of quantum field theory, string theory, and speculative quantum gravity theories.

History

The history of quantum mechanics dates back to the 1838 discovery of cathode rays by Michael Faraday. This was
followed by the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by
Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of
Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta", or "energy elements",
precisely matched the observed patterns of black body radiation. According to Planck, each energy element E is
proportional to its frequency v:

E = hv

where h is Planck's constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption
and emission of radiation and had nothing to do with the physical reality of the radiation itself. However, in 1905
Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in
which shining light on certain materials can eject electrons from the material. Einstein postulated that light itself
consists of individual quanta of energy, later called photons.

The foundations of quantum mechanics were established during the first half of the twentieth century by Niels Bohr,
Werner Heisenberg, Max Planck, Louis de Broglie, Albert Einstein, Erwin Schrodinger, Max Born, John von
Neumann, Paul Dirac, Wolfgang Pauli, David Hilbert, and others. In the mid-1920s, developments in quantum
mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and
Heisenberg published results that closed the "Old Quantum Theory". Out of deference to their dual state as particles,
light quanta came to be called photons (1926). From Einstein's simple postulation was born a flurry of debating,

Quantum Dynamics 7 1

theorizing and testing. Thus the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth
Solvay Conference in 1927.

The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it
was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or
quanta, Albert Einstein further developed this idea to show that an electromagnetic wave such as light could be
described as a particle - later called the photon - with a discrete energy that was dependent on its frequency. This led
to a theory of unity between subatomic particles and electromagnetic waves called wave— particle duality in which
particles and waves were neither one nor the other, but had certain properties of both.

While quantum mechanics traditionally described the world of the very small, it is also needed to explain certain

recently investigated macroscopic systems such as superconductors and superfluids.

mi
The word quantum derives from Latin, meaning "how great" or "how much". In quantum mechanics, it refers to a

discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see

Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of

physics dealing with atomic and sub-atomic systems which is today called quantum mechanics. It is the underlying

mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state

physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry,

particle physics, nuclear chemistry, and nuclear physics. Some fundamental aspects of the theory are still actively

studied. [6]

Quantum mechanics is essential to understand the behavior of systems at atomic length scales and smaller. For
example, if classical mechanics governed the workings of an atom, electrons would rapidly travel towards and
collide with the nucleus, making stable atoms impossible. However, in the natural world the electrons normally
remain in an uncertain, non-deterministic "smeared" (wave— particle wave function) orbital path around or through
the nucleus, defying classical electromagnetism.

Quantum mechanics was initially developed to provide a better explanation of the atom, especially the differences in
the spectra of light emitted by different isotopes of the same element. The quantum theory of the atom was
developed as an explanation for the electron remaining in its orbit, which could not be explained by Newton's laws
of motion and Maxwell's laws of classical electromagnetism.

Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot
account:

• The quantization of certain physical properties

• Wave— particle duality

• The uncertainty principle

• Quantum entanglement

Mathematical formulations

ro]

In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac and John von
Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state
vectors"). Formally, these reside in a complex separable Hilbert space (variously called the "state space" or the
"associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other
words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective
space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and
momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is
just the product of two complex planes. Each observable is represented by a maximally Hermitian (precisely: by a
self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector
of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the

Quantum Dynamics 72

operator's spectrum is discrete, the observable can only attain those discrete eigenvalues.

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave
function, also referred to as state vector in a complex vector space. This abstract mathematical object allows for
the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the
probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical
mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum,
with accuracy. For instance, electrons may be considered to be located somewhere within a region of space, but with
their exact positions being unknown. Contours of constant probability, often referred to as "clouds", may be drawn
around the nucleus of an atom to conceptualize where the electron might be located with the most probability.
Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate
momentum.

According to one interpretation, as the result of a measurement the wave function containing the probability
information for a system collapses from a given initial state to a particular eigenstate. The possible results of a
measurement are the eigenvalues of the operator representing the observable — which explains the choice of
Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable
in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty
principle is represented by the statement that the operators corresponding to certain observables do not commute.

The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most
difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in
which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the
decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been
extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the
concept of "wavefunction collapse"; see, for example, the relative state interpretation. The basic idea is that when a
quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the

original quantum system ceases to exist as an independent entity. For details, see the article on measurement in

ri2i
quantum mechanics. Generally, quantum mechanics does not assign definite values. Instead, it makes predictions

using probability distributions; that is, it describes the probability of obtaining possible outcomes from measuring an

ri3i

observable. Often these results are skewed by many causes, such as dense probability clouds or quantum state
nuclear attraction. Naturally, these probabilities will depend on the quantum state at the "instant" of the

measurement. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with
a definite value of a particular observable. These are known as eigenstates of the observable ("eigen" can be
translated from German as meaning inherent or characteristic).

In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate.
Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of
occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum
(since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it only provides a
range of probabilities of where that particle might be given its momentum and momentum probability. Therefore, it
is helpful to use different words to describe states having uncertain values and states having definite values
(eigenstate). Usually, a system will not be in an eigenstate of the observable (particle) we are interested in. However,
if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalised eigenstate) of

ri7i

that observable. This process is known as wavefunction collapse, a controversial and much debated process. It
involves expanding the system under study to include the measurement device. If one knows the corresponding wave
function at the instant before the measurement, one will be able to compute the probability of collapsing into each of
the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that
is a wave packet centered around some mean position x , neither an eigenstate of position nor of momentum. When
one measures the position of the particle, it is impossible to predict with certainty the result. It is probable, but not

Quantum Dynamics

certain, that it will be near x , where the amplitude of the wave function is large. After the measurement is

[181

performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.

The time evolution of a quantum state is described by the Schrodinger equation, in which the Hamiltonian (the
operator corresponding to the total energy of the system) generates time evolution. The time evolution of wave
functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of

ri9i

what the wavefunction will be at any later time.

During a measurement, on the other hand, the change of the wavefunction into another one is not deterministic; it is
unpredictable, i.e. random. A time-evolution simulation can be seen here. Wave functions can change as time

progresses. An equation known as the Schrodinger equation describes how wavefunctions change in time, a role
similar to Newton's second law in classical mechanics. The Schrodinger equation, applied to the aforementioned
example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity,
like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses,
which means that the position becomes more uncertain. This also has the effect of turning position eigenstates

(which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position

* * [22]
eigenstates.

Some wave functions produce probability
distributions that are constant, or independent of
time, such as when in a stationary state of
constant energy, time drops out of the absolute
square of the wave function. Many systems that
are treated dynamically in classical mechanics are
described by such "static" wave functions. For
example, a single electron in an unexcited atom is
pictured classically as a particle moving in a
circular trajectory around the atomic nucleus,
whereas in quantum mechanics it is described by
a static, spherically symmetric wavefunction
surrounding the nucleus (Fig. 1). (Note that only
the lowest angular momentum states, labeled s,

are spherically symmetric)

[23]

^^^^^^E^^^^^l

^^^^H^34

Fig. 1 : Probability densities corresponding to the wavefunctions of an
electron in a hydrogen atom possessing definite energy levels (increasing

from the top of the image to the bottom: n = 1, 2, 3, ...) and angular
momentum (increasing across from left to right: s, p, d, ...). Brighter areas

correspond to higher probability density in a position measurement.

Wavefunctions like these are directly comparable to Chladni's figures of

acoustic modes of vibration in classical physics and are indeed modes of

oscillation as well: they possess a sharp energy and thus a keen frequency.

The angular momentum and energy are quantized, and only take on discrete

values like those shown (as is the case for resonant frequencies in

acoustics).

The Schrodinger equation acts on the entire

probability amplitude, not merely its absolute

value. Whereas the absolute value of the

probability amplitude encodes information about

probabilities, its phase encodes information about

the interference between quantum states. This

gives rise to the wave-like behavior of quantum

states. It turns out that analytic solutions of

Schrodinger's equation are only available for a

small number of model Hamiltonians, of which

the quantum harmonic oscillator, the particle in a

box, the hydrogen molecular ion and the

hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron

than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating

approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a
simple quantum mechanical model to generate results for a more complicated model related to the simple model by,

Quantum Dynamics 74

for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion"
approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior.
The deviations can be calculated based on the classical motion. This approach is important for the field of quantum
chaos.

There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most
commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac,
which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by
Werner Heisenberg) and wave mechanics (invented by Erwin Schrodinger). In this formulation, the

instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables".

Examples of observables include energy, position, momentum, and angular momentum. Observables can be either

T271
continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).

An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a

quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the

quantum-mechanical counterpart of action principles in classical mechanics.

Interactions with other scientific theories

The rules of quantum mechanics are fundamental; they assert that the state space of a system is a Hilbert space and
that observables of that system are Hermitian operators acting on that space; they do not tell us which Hilbert space
or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum
system. An important guide for making these choices is the correspondence principle, which states that the
predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or,
equivalently, larger quantum numbers (i.e. whereas a single particle exhibits a degree of randomness, in systems
incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of
random behaviour approaches zero). In other words, classical mechanics is simply a quantum mechanics of large
systems. This "high energy" limit is known as the classical or correspondence limit. One can even start from an
established classical model of a particular system, and attempt to guess the underlying quantum model that would
give rise to the classical model in the correspondence limit.

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was
non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an
explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the
classical harmonic oscillator.

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrodinger
equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories
were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from
their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the
development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The
first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the
electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing
electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat
charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example,
the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a
classical -e 2 /(Aiv e r) Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the
electromagnetic field play an important role, such as in the emission of photons by charged particles.
Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum
field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of
subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their

Quantum Dynamics 75

quantized forms, into a single quantum field theory known as electroweak theory, by the physicists Abdus Salam,

— ro on

Sheldon Glashow and Steven Weinberg. These three men shared the Nobel Prize in Physics in 1979 for this work.

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical
approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a
complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most
accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The
resolution of these incompatibilities is an area of active research, and theories such as string theory are among the
possible candidates for a future theory of quantum gravity.

Classical mechanics has been extended into the complex domain, and complex classical mechanics exhibits
behaviours similar to quantum mechanics.

Quantum mechanics and classical physics

Predictions of quantum mechanics have been verified experimentally to a extremely high degree of accuracy.
According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of
quantum mechanics, and classical mechanics is just an approximation for large systems (or a statistical quantum
mechanics of a large collection of particles). The laws of classical mechanics thus follow from the laws of quantum
mechanics as a statistical average at the limit of large systems or large quantum numbers. However, chaotic
systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and
quantum descriptions in these systems.

Quantum coherence is an essential difference between classical and quantum theories, and is illustrated by the
Einstein-Podolsky-Rosen paradox. Quantum interference involves adding together probability amplitudes, whereas
when classical waves interfere there is an adding together of intensities. For microscopic bodies, the extension of the
system is much smaller than the coherence length, which gives rise to long-range entanglement and other nonlocal
phenomena characteristic of quantum systems. Quantum coherence is not typically evident at macroscopic scales,
although an exception to this rule can occur at extremely low temperatures, when quantum behavior can manifest
itself on more macroscopic scales (see Bose-Einstein condensate and Quantum machine). This is in accordance with
the following observations:

• Many macroscopic properties of a classical system are a direct consequences of the quantum behavior of its parts.
For example, the stability of bulk matter (which consists of atoms and molecules which would quickly collapse
under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic
properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.

• While the seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more
apparent when dealing with extremely fast-moving or extremely tiny particles, the laws of classical Newtonian

physics remain accurate in predicting the behavior of the vast majority of large objects — of the order of the size of

T331
large molecules and bigger — at velocities much smaller than the velocity of light.

Quantum Dynamics 76

Relativity and quantum mechanics

Main articles: Quantum gravity and Theory of everything

Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being
indisputably supported by rigorous and repeated empirical evidence and while they do not directly contradict each
other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one
cohesive model.

Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to
the field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as
the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of
space at one time. He also was the first to notice some of the apparently exotic consequences of entanglement and
used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics had
unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that, although
Einstein was correct in identifying seemingly paradoxical implications of quantum mechanical nonlocality, these
implications could be experimentally tested. Alain Aspect's initial experiments in 1982, and many subsequent
experiments since, have verified quantum entanglement.

According to the paper of J. Bell and the Copenhagen interpretation (the common interpretation of quantum
mechanics by physicists since 1927), and contrary to Einstein's ideas, quantum mechanics was not at the same time

• a "realistic" theory

• and a local theory.

The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the
state of one particle and instantaneously change the state of its entangled partner, although the two particles can be
an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens.
Quantum entanglement is at the basis of quantum cryptography, with high-security commercial applications in
banking and government.

Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum
mechanics is not an urgent issue in those applications. However, the lack of a correct theory of quantum gravity is an
important issue in cosmology and physicists' search for an elegant "theory of everything". Thus, resolving the
inconsistencies between both theories has been a major goal of twentieth- and twenty-first-century physics. Many
prominent physicists, including Stephen Hawking, have labored in the attempt to discover a theory underlying
everything, combining not only different models of subatomic physics, but also deriving the universe's four
forces — the strong force, electromagnetism, weak force, and gravity — from a single force or phenomenon. While
Stephen Hawking was initially a believer in the Theory of Everything, after considering Godel's Incompleteness

Theorem, concluded that one was not obtainable, and stated such publicly in his lecture, "Godel and the end of

T351
physics" in 2002. One of the leaders in this field is Edward Witten, a theoretical physicist who formulated the

groundbreaking M-theory, which is an attempt at describing the supersymmetrical based string theory.

Attempts at a unified field theory

As of 2011 the quest for unifying the fundamental forces through quantum mechanics is still ongoing. Quantum
electrodynamics (or "quantum electromagnetism"), which is currently (in the perturbative regime at least) the most
accurately tested physical theory, has been successfully merged with the weak nuclear force into the electroweak
force and work is currently being done to merge the electroweak and strong force into the electrostrong force.

Current predictions state that at around 10 GeV the three aforementioned forces are fused into a single unified

[371
field, Beyond this "grand unification," it is speculated that it may be possible to merge gravity with the other

19
three gauge symmetries, expected to occur at roughly 10 GeV. However — and while special relativity is

parsimoniously incorporated into quantum electrodynamics — the expanded general relativity, currently the best

Quantum Dynamics 77

theory describing the gravitation force, has not been fully incorporated into quantum theory.

Philosophical implications

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical
debate and many interpretations. Even fundamental issues such as Max Born's basic rules concerning probability
amplitudes and probability distributions took decades to be appreciated.

R81
Richard Feynman said, "I think I can safely say that nobody understands quantum mechanics."

The Copenhagen interpretation, due largely to the Danish theoretical physicist Niels Bohr, is the interpretation of the
quantum mechanical formalism most widely accepted amongst physicists. According to it, the probabilistic nature of
quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but
instead must be considered to be a final renunciation of the classical ideal of causality. In this interpretation, it is
believed that any well-defined application of the quantum mechanical formalism must always make reference to the
experimental arrangement, due to the complementarity nature of evidence obtained under different experimental
situations.

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. (A

view paraphrased as "God does not play dice with the universe.") Einstein held that there should be a local hidden

variable theory underlying quantum mechanics and that, consequently, the present theory was incomplete. He

produced a series of objections to the theory, the most famous of which has become known as the

Einstein-Podolsky-Rosen paradox. John Bell showed that the EPR paradox led to experimentally testable differences

between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy

[39]
of quantum mechanics, thus demonstrating that the physical world cannot be described by local realistic theories.

The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point

of view.

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum
theory simultaneously occur in a multiverse composed of mostly independent parallel universes. This is not
accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of
the collapse of the wave packet: All the possible consistent states of the measured system and the measuring
apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other
interpretations) quantum superposition. Such a superposition of consistent state combinations of different systems is
called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior governed by
probabilities, because we can observe only the universe, i.e. the consistent state contribution to the mentioned
superposition, we inhabit. Everett's interpretation is perfectly consistent with John Bell's experiments and makes
them intuitively understandable. However, according to the theory of quantum decoherence, the parallel universes
will never be accessible to us. This inaccessibility can be understood as follows: Once a measurement is done, the
measured system becomes entangled with both the physicist who measured it and a huge number of other particles,
some of which are photons flying away towards the other end of the universe; in order to prove that the wave
function did not collapse one would have to bring all these particles back and measure them again, together with the
system that was measured originally. This is completely impractical, but even if one could theoretically do this, it
would destroy any evidence that the original measurement took place (including the physicist's memory).

Quantum Dynamics 78

Applications

Quantum mechanics had enormous success in explaining many of the features of our world. The individual
behaviour of the subatomic particles that make up all forms of matter — electrons, protons, neutrons, photons and
others — can often only be satisfactorily described using quantum mechanics. Quantum mechanics has strongly
influenced string theory, a candidate for a theory of everything (see reductionism) and the multiverse hypothesis.

Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or
molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. (Relativistic)
quantum mechanics can in principle mathematically describe most of chemistry. Quantum mechanics can provide
quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are

energetically favorable to which others, and by approximately how much. Most of the calculations performed in

T421
computational chemistry rely on quantum mechanics.

Much of modern technology operates at a scale where quantum effects are significant. Examples include the laser,
the transistor (and thus the microchip), the electron microscope, and magnetic resonance imaging. The study of
semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to
develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal
is the development of quantum computers, which are expected to perform certain computational tasks exponentially
faster than classical computers. Another active research topic is quantum teleportation, which deals with techniques
to transmit quantum information over arbitrary distances.

Quantum tunneling is vital in many devices, even in the simple light switch, as otherwise the electrons in the electric
current could not penetrate the potential barrier made up of a layer of oxide. Flash memory chips found in USB
drives use quantum tunneling to erase their memory cells.

Quantum mechanics primarily applies to the atomic regimes of matter and energy, but some systems exhibit
quantum mechanical effects on a large scale; superfluidity (the frictionless flow of a liquid at temperatures near
absolute zero) is one well-known example. Quantum theory also provides accurate descriptions for many previously

unexplained phenomena such as black body radiation and the stability of electron orbitals. It has also given insight

T431
into the workings of many different biological systems, including smell receptors and protein structures. Recent

work on photosynthesis has provided evidence that quantum correlations play an essential role in this most

T441
fundamental process of the plant kingdom. Even so, classical physics often can be a good approximation to

results otherwise obtained by quantum physics, typically in circumstances with large numbers of particles or large

quantum numbers. (However, some open questions remain in the field of quantum chaos.)

Examples
Free particle

For example, consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the
particle can be described as the properties of a wave. Therefore, its quantum state can be represented as a wave of
arbitrary shape and extending over space as a wave function. The position and momentum of the particle are
observables. The Uncertainty Principle states that both the position and the momentum cannot simultaneously be
measured with full precision at the same time. However, one can measure the position alone of a moving free
particle creating an eigenstate of position with a wavefunction that is very large (a Dirac delta) at a particular
position x and zero everywhere else. If one performs a position measurement on such a wavefunction, the result x
will be obtained with 100% probability (full certainty). This is called an eigenstate of position (mathematically more
precise: a generalized position eigenstate (eigendistribution)). If the particle is in an eigenstate of position then its
momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum then its

Quantum Dynamics

T451
position is completely unknown. In an eigenstate of momentum having a plane wave form, it can be shown that

the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate.

Step potential

The potential in this case is given by:

The step potential with incident and exiting
waves shown.

V(x)

0, x < 0,
Vo, x>0,

The solutions are superpositions of left and right moving waves:

ip L (x) = -j= (A r e ikox + Ae'^ *) x<0,
V fco v J

ij> R (x) = 4= (B T e iklx + B ie - ik1 *) x>0

where the wave vectors are related to the energy via

fc = \j2mEjk 1 ' and
Jfei = ^2m(E - V )/h 2

and the coefficients A and B are determined from the boundary conditions and by imposing a continuous derivative
to the solution.

Each term of the solution can be interpreted as an incident, reflected of transmitted component of the wave, allowing
the calculation of transmission and reflection coefficients. In contrast to classical mechanics, incident particles with
energies higher than the size of the potential step are still partially reflected.

Quantum Dynamics

Rectangular potential barrier

This is a model for the quantum tunneling effect, which has important applications to modern devices such as flash
memory and the scanning tunneling microscope.

Particle in a box

The particle in a 1 -dimensional potential energy box is the most simple
example where restraints lead to the quantization of energy levels. The
box is defined as having zero potential energy inside a certain region
and infinite potential energy everywhere outside that region. For the

1 -dimensional case in the a; direction, the time-independent

1471
Schrodinger equation can be written as:

v=o

1 -dimensional potential energy box (or infinite
potential well)

h 2 d 2 ip

= Eijj.

2m dx 2
Writing the differential operator

p x = -in—
dx

the previous equation can be seen to be evocative of the classic analogue

— f=E
2m Fx

with ^ as the energy for the state ip , in this case coinciding with the kinetic energy of the particle.
The general solutions of the Schrodinger equation for the particle in a box are:

h 2 k 2

iP(x) = Ae ikx + Be~ ikx

or, from Euler's formula,

tp(x) = C sin kx + D cos kx.
The presence of the walls of the box determines the values of C, D, and k. At each wall (x = and x — L), ip - 0.
Thus when x = 0,

t/,(0) =0 = CsinO + DcosO = D

and so D = 0. When x = L,

^(Z ; ) = = CsinfcL.

C cannot be zero, since this would conflict with the Born interpretation. Therefore sin kL = 0, and so it must be that

kL is an integer multiple of Jt. Therefore,

TlTT
k = —
L

The quantization of energy levels follows from this constraint on k, since

1,2,3,

fc2 2 2 2l2

a it n n h

2mL 2 8mL 2 '

Quantum Dynamics 8 1

Finite potential well

This is generalization of the infinite potential well problem to potential wells of finite depth.

Harmonic oscillator

As in the classical case, the potential for the quantum harmonic oscillator is given by:

V{x) = —mw x

This problem can be solved either by directly solving the Schrodinger equation directly, which is not trivial, or by
using the more elegant ladder method, first proposed by Paul Dirac. The eigenstates are given by:

1 fmwV^

where H are the Hermite polynomials:

JTJ

and the corresponding energy levels are

E n = huj ln+ -
This is another example which illustrates the quantification of energy for bound states.

Notes

[I] J. Mehra and H. Rechenberg, The historical development of quantum theory, Springer- Verlag, 1982.

[2] T.S. Kuhn, Black-body theory and the quantum discontinuity 1894-1912, Clarendon Press, Oxford, 1978.

[3] A. Einstein, Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (On a heuristic point of view

concerning the production and transformation of light), Annalen der Physik 17 (1905) 132-148 (reprinted in The collected papers of Albert

Einstein, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149-166, in German; see also Einstein's early work on the

quantum hypothesis, ibid. pp. 134-148).
[4] "Merriam-Webster.com" (http://www.merriam-webster.com/dictionary/quantum). Merriam-Webster.com. 2010-08-13. . Retrieved

2010-10-15.
[5] Edwin Thall. "FCCJ.org" (http://mooni.fccj.org/~ethall/quantum/quant.htm). Mooni.fccj.org. . Retrieved 2010-10-15.
[6] Compare the list of conferences presented here (http://ysfine.com/).

[7] Oocities.com (http://web.archive.Org/web/20091026095410/http://geocities.com/mik_malm/quantmech.html)
[8] P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1930.
[9] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 (English translation: Mathematical Foundations

of Quantum Mechanics, Princeton University Press, 1955).
[10] Greiner, Walter; Muller, Berndt (1994). Quantum Mechanics Symmetries, Second edition (http://books.google.com/

books?id=gCfvWx6vuzUC&pg=PA52). Springer-Verlag. p. 52. ISBN 3-540-58080-8. .,

[II] "AIP.org" (http://www.aip.org/history/heisenberg/p08a.htm). AIP.org. . Retrieved 2010-10-15.

[12] Greenstein, George; Zajonc, Arthur (2006). The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Second
edition (http://books.google.com/books?id=5t0tm0FBlCsC&pg=PA215). Jones and Bartlett Publishers, Inc. p. 215. ISBN 0-7637-2470-X.

[13] probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave

function eigenvalue, such that the probability is the squared modulus of the complex amplitude
[14] "Actapress.com" (http://www.actapress.com/PaperInfo.aspx?PaperID=25988&reason=500). Actapress.com. . Retrieved 2010-10-15.
[15] Hirshleifer, Jack (2001). The Dark Side of the Force: Economic Foundations of Conflict Theory (http://books.google.com/

books?id=W2J2IXgiZVgC&pg=PA265). Campbridge University Press, p. 265. ISBN 0-521-80412-4. .,
[16] Dict.cc (http://www.dict.cc/german-english/eigen.html)

De.pons.eu (http://de.pons.eu/deutsch-englisch/eigen)
[17] "PHY.olemiss.edu" (http://www.phy.olemiss.edu/~luca/Topics/qm/collapse.html). PHY.olemiss.edu. 2010-08-16. . Retrieved

2010-10-15.
[18] "Farside.ph.utexas.edu" (http://farside.ph.utexas.edu/teaching/qmech/lectures/node28.html). Farside.ph.utexas.edu. . Retrieved

2010-10-15.

Quantum Dynamics 82

[19] "Reddit.com" (http://www.reddit.eom/r/philosophy/comments/8p2qv/determinism_and_naive_realism/). Reddit.com. 2009-06-01. .

Retrieved 2010-10-15.
[20] Michael Trott. "Time-Evolution of a Wavepacket in a Square Well — Wolfram Demonstrations Project" (http://demonstrations. wolfram.

com/TimeEvolutionOfAWavepacketlnASquareWell/). Demonstrations.wolfram.com. . Retrieved 2010-10-15.
[21] Michael Trott. "Time Evolution of a Wavepacket In a Square Well" (http://demonstrations.wolfram.com/

TimeEvolutionOfAWavepacketlnASquareWell/). Demonstrations.wolfram.com. . Retrieved 2010-10-15.
[22] Mathews, Piravonu Mathews; Venkatesan, K. (1976). A Textbook of Quantum Mechanics (http://books.google.com/

books?id=_qzslDD3TcsC&pg=PA36). Tata McGraw-Hill. p. 36. ISBN 0-07-096510-2. .,
[23] "Wave Functions and the Schrodinger Equation" (http://physics.ukzn.ac.za/~petruccione/Physl20/Wave Functions and the Schrodinger

Equation.pdf) (PDF). . Retrieved 2010-10-15.
[24] "Spaceandmotion.com" (http://www.spaceandmotion.com/physics-quantum-mechanics-werner-heisenberg.htm). Spaceandmotion.com. .

Retrieved 2010-10-15.
[25] Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max

Born has been obfuscated. A 2005 biography of Born details his role as the creator of the matrix formulation of quantum mechanics. This was

recognized in a paper by Heisenberg, in 1940, honoring Max Planck. See: Nancy Thorndike Greenspan, "The End of the Certain World: The

Life and Science of Max Born" (Basic Books, 2005), pp. 124 - 128, and 285 - 286.
[26] "IF.uj.edu.pl" (http://th-www.if.uj.edu.pl/acta/voll9/pdf/vl9p0683.pdf) (PDF). . Retrieved 2010-10-15.
[27] "OCW.ssu.edu" (http://ocw.usu.edu/physics/classical-mechanics/pdf_lectures/06.pdf) (PDF). . Retrieved 2010-10-15.
[28] "The Nobel Prize in Physics 1979" (http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html). Nobel Foundation. .

Retrieved 2010-02-16.
[29] Complex Elliptic Pendulum (http://arxiv.org/abs/1001.0131), Carl M. Bender, Daniel W. Hook, Karta Kooner
[30] "Scribd.com" (http://www.scribd.com/doc/5998949/Quantum-mechanics-course-iwhatisquantummechanics). Scribd.com. 2008-09-14. .

Retrieved 2010-10-15.
[31] Philsci-archive.pitt.edu (http://philsci-archive.pitt.edu/archive/00002328/01/handbook.pdf)
[32] "Academic.brooklyn.cuny.edu" (http://academic.brooklyn.cuny.edu/physics/sobel/Nucphys/atomprop.html).

Academic.brooklyn.cuny.edu. . Retrieved 2010-10-15.
[33] "Cambridge.org" (http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf) (PDF). . Retrieved 2010-10-15.
[34] "There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4. — V. B. Berestetskii, E. M. Lifshitz, L P

Pitaevskii (1971). J. B. Sykes, J. S. Bell (translators). Relativistic Quantum Theory 4, part I. Course of Theoretical Physics (Landau and

Lifshitz) ISBN 0080160255
[35] http://www.damtp.cam.ac.uk/strings02/dirac/hawking/
[36] "Life on the lattice: The most accurate theory we have" (http://latticeqcd.blogspot.com/2005/06/most-accurate-theory-we-have.html).

Latticeqcd.blogspot.com. 2005-06-03. . Retrieved 2010-10-15.
[37] Parker, B. (1993). Overcoming some of the problems, pp. 259—279.

[38] The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003) by Tony Hey and Patrick Walters
[39] "Plato.stanford.edu" (http://plato.stanford.edu/entries/qm-action-distance/). Plato.stanford.edu. 2007-01-26. . Retrieved 2010-10-15.
[40] "Plato.stanford.edu" (http://plato.stanford.edu/entries/qm-everett/). Plato.stanford.edu. . Retrieved 2010-10-15.
[41] Books.google.com (http://books. google. com/books ?id=vdXU6SD4_UYC). Books.google.com. . Retrieved 2010-10-23.
[42] "en.wikiboos.org" (http://en.wikibooks.org/wiki/Computational_chemistry/Applications_of_molecular_quantum_mechanics).

En.wikibooks.org. . Retrieved 2010-10-23.
[43] Anderson, Mark (2009-01-13). "Discovermagazine.com" (http://discovermagazine.com/2009/feb/

13-is-quantum-mechanics-controlling-your-thoughts/article_view?b_start:int=l&-C). Discovermagazine.com. . Retrieved 2010-10-23.
[44] "Quantum mechanics boosts photosynthesis" (http://physicsworld.com/cws/article/news/41632). physicsworld.com. . Retrieved

2010-10-23.
[45] Davies, P. C. W.; Betts, David S. (1984). Quantum Mechanics, Second edition (http://books.google.com/books?id=XRyHCrGNstoC&

pg=PA79). Chapman and Hall. p. 79. ISBN 0-7487-4446-0. .,
[46] Books.Google.com (http://books.google.com/books?id=tKm-Ekwke_UC). Books.Google.com. 2007-08-30. . Retrieved 2010-10-23.
[47] Derivation of particle in a box, chemistry.tidalswan.com (http://chemistry.tidalswan. com/index. php?title=Quantum_Mechanics)

Quantum Dynamics 83

References

The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a
minimum of technical apparatus.

Chester, Marvin (1987) Primer of Quantum Mechanics. John Wiley. ISBN 0-486-42878-8

Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN

0-691-08388-6. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing

many insights for the expert.

Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The

most technical of the works cited here. Passages using algebra, trigonometry, and bra-ket notation can be passed

over on a first reading.

N. David Mermin, 1990, "Spooky actions at a distance: mysteries of the QT" in his Boojums all the way through.

Cambridge University Press: 110-76.

Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus

Books. Chpts. 5-8. Includes cosmological and philosophical considerations.

More technical:

Bryce DeWitt, R. Neill Graham, eds., 1973. The Many-Worlds Interpretation of Quantum Mechanics, Princeton

Series in Physics, Princeton University Press. ISBN 0-691-08 13 1-X

Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. ISBN 01985201 15. The beginning chapters make

up a very clear and comprehensible introduction.

Hugh Everett, 1957, "Relative State Formulation of Quantum Mechanics," Reviews of Modern Physics 29:

454-62.

Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. 1-3.

Addison-Wesley. ISBN 0738200085.

Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-1 1 1892-7.

OCLC 40251748. A standard undergraduate text.

Max Jammer, 1966. The Conceptual Development of Quantum Mechanics. McGraw Hill.

Hagen Kleinert, 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,

3rd ed. Singapore: World Scientific. Draft of 4th edition, (http://www.physik.fu-berlin.de/~kleinert/b5)

Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon Press. ISBN 0-08-203204-1

George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN

0-486-43517-2.

Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G M. Temmer. North

Holland, John Wiley & Sons. Cf. chpt. IV, section III.

Omnes, Roland (1999). Understanding Quantum Mechanics . Princeton University Press. ISBN 0-691-00435-8.

OCLC 39849482.

Scerri, Eric R., 2006. The Periodic Table: Its Story and Its Significance. Oxford University Press. Considers the

extent to which chemistry and the periodic system have been reduced to quantum mechanics. ISBN

0-19-530573-6

Transnational College of Lex (1996). What is Quantum Mechanics? A Physics Adventure. Language Research

Foundation, Boston. ISBN 0-9643504-1-6. OCLC 34661512.

von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press.

ISBN 0691028931.

Hermann Weyl, 1950. The Theory of Groups and Quantum Mechanics, Dover Publications.

D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. Compendium of quantum physics, Concepts, experiments,

history and philosophy, Springer- Verlag, Berlin, Heidelberg.

Quantum Dynamics 84

Further reading

• Bernstein, Jeremy (2009). Quantum Leaps (http://books. google. com/books ?id=j0Me3brYOL0C&
printsec=frontcover). Cambridge, Massachusetts: Belknap Press of Harvard University Press.
ISBN 9780674035416.

• Bohm, David (1989). Quantum Theory. Dover Publications. ISBN 0-486-65969-0.

• Eisberg, Robert; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
(2nd ed.). Wiley. ISBN 0-471-87373-X.

• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.

• Merzbacher, Eugen (1998). Quantum Mechanics. Wiley, John & Sons, Inc. ISBN 0-471-88702-1.

• Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0-201-53929-2.

• Shankar, R. (1994). Principles of Quantum Mechanics. Springer. ISBN 0-306-44790-8.

External links

• A foundation approach to quantum Theory that does not rely on wave-particle duality. (http://www.mesacc.
edu/~kevinlg/i256/QM_basics.pdf)

• The Modern Revolution in Physics (http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html) - an
online textbook.

• J. O'Connor and E. F. Robertson: A history of quantum mechanics, (http://www-history.mcs.st-andrews.ac.uk/
history/HistTopics/The_Quantum_age_begins.html)

• Introduction to Quantum Theory at Quantiki. (http://www.quantiki.org/wiki/index.php/
Introduction_to_Quantum_Theory)

• Quantum Physics Made Relatively Simple (http://bethe.cornell.edu/): three video lectures by Hans Bethe

• H is for h-bar. (http://www.nonlocal.com/hbar/)

• Quantum Mechanics Books Collection (http://www.freebookcentre.net/Physics/Quantum-Mechanics-Books.
html): Collection of free books

Course material

• Doron Cohen: Lecture notes in Quantum Mechanics (comprehensive, with advanced topics), (http://arxiv.org/
abs/quant-ph/0605180)

• MIT OpenCourseWare: Chemistry (http://ocw.mit.edu/OcwWeb/Chemistry/index.htm).

• MIT OpenCourseWare: Physics (http://ocw.mit.edu/OcwWeb/Physics/index.htm). See 8.04 (http://ocw.
mit.edu/OcwWeb/Physics/8-04Spring-2006/CourseHome/index.htm)

• Stanford Continuing Education PHY 25: Quantum Mechanics (http://www.youtube.eom/stanford#g/c/
84C10A9CB1D13841) by Leonard Susskind, see course description (http://continuingstudies.stanford.edu/
courses/course.php?cid=20072_PHY 25) Fall 2007

• 5Vi Examples in Quantum Mechanics (http://www.physics.csbsju.edu/QM/)

• Imperial College Quantum Mechanics Course, (http://www.imperial.ac.uk/quantuminformation/qi/tutorials)

• Spark Notes - Quantum Physics, (http://www.sparknotes.com/testprep/books/sat2/physics/
chapter 1 9section3 .rhtml)

• Quantum Physics Online : interactive introduction to quantum mechanics (RS applets). (http://www.
quantum-physics . poly technique . fr/)

• Experiments to the foundations of quantum physics with single photons. (http://www.didaktik.physik.
uni-erlangen.de/quantumlab/english/index.html)

• Motion Mountain, Volume IV (http://www.motionmountain.net/download.html) - A modern introduction to
quantum theory, with several animations.

• AQME (http://www.nanohub.org/topics/AQME) : Advancing Quantum Mechanics for Engineers — by
T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on nanohub

Quantum Dynamics 85

• Quantum Mechanics (http://www.lsr.ph.ic.ac.uk/~plenio/lecture.pdf) by Martin Plenio

• Quantum Mechanics (http://farside.ph.utexas.edu/teaching/qm/389.pdf) by Richard Fitzpatrick

• Online course on Quantum Transport (http://nanohub.org/resources/2039)

FAQs

• Many-worlds or relative-state interpretation, (http://www.hedweb.com/manworld.htm)

• Measurement in Quantum mechanics, (http://www.mtnmath.com/faq/meas-qm.html)

Media

• Lectures on Quantum Mechanics by Leonard Susskind (http://www.youtube.com/
view_play_list?p=84C10A9CBlD13841)

• Everything you wanted to know about the quantum world (http://www.newscientist.com/channel/
fundamentals/quantum-world) — archive of articles from New Scientist.

• Quantum Physics Research (http://www.sciencedaily.com/news/matter_energy/quantum_physics/) from
Science Daily

• Overbye, Dennis (December 27, 2005). "Quantum Trickery: Testing Einstein's Strangest Theory" (http://www.
nytimes.com/2005/12/27/science/27eins.html?ex=1293339600&en=caf5d835203c3500&ei=5090). The
New York Times. Retrieved April 12, 2010.

• Audio: Astronomy Cast (http://www.astronomycast.com/physics/ep-138-quantum-mechanics/) Quantum
Mechanics — June 2009. Fraser Cain interviews Pamela L. Gay.

Philosophy

• "Quantum Mechanics" (http://plato.stanford.edu/entries/qm) entry by Jenann Ismael. in the Stanford
Encyclopedia of Philosophy

• "Measurement in Quantum Theory" (http://plato.stanford.edu/entries/qm) entry by Henry Krips. in the
Stanford Encyclopedia of Philosophy

Mathematical Formulations of Quantum Dynamics 86

Mathematical Formulations of Quantum
Dynamics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous
description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed
prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces
and operators on these spaces. Many of these structures are drawn from functional analysis, a research area within
pure mathematics that was influenced in part by the needs of quantum mechanics. In brief, values of physical
observables such as energy and momentum were no longer considered as values of functions on phase space, but as
eigenvalues; more precisely: as spectral values (point spectrum plus absolute continuous plus singular continuous
spectrum) of linear operators in Hilbert space.

These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of
quantum state and quantum observable which are radically different from those used in previous models of physical
reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a
definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by
Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the
non-commutativity of quantum observables.

Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly
of differential geometry and partial differential equations; probability theory was used in statistical mechanics.
Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were
formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between
1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists
continued to think of quantum theory within the confines of what is now called classical physics, and in particular
within the same mathematical structures. The most sophisticated example of this is the
Sommerfeld— Wilson— Ishiwara quantization rule, which was formulated entirely on the classical phase space.

History of the formalism

The "old quantum theory" and the need for new mathematics

In the 1890s, Planck was able to derive the blackbody spectrum which was later used to avoid the classical
ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of radiation with matter, energy
could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between
the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called
Planck's constant in his honor.

In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were
actual particles, which were later dubbed photons.

Mathematical Formulations of Quantum Dynamics

All of these developments were phenomenological and flew in the face
of the theoretical physics of the time. Bohr and Sommerfeld went on to
modify classical mechanics in an attempt to deduce the Bohr model
from first principles. They proposed that, of all closed classical orbits
traced by a mechanical system in its phase space, only the ones that
enclosed an area which was a multiple of Planck's constant were
actually allowed. The most sophisticated version of this formalism was
the so-called Sommerfeld— Wilson— Ishiwara quantization. Although
the Bohr model of the hydrogen atom could be explained in this way,
the spectrum of the helium atom (classically an unsolvable 3-body
problem) could not be predicted. The mathematical status of quantum
theory remained uncertain for some time.

Increasing energy

of orbits

A photon is emitted
with energy E =hf

In 1923 de Broglie proposed that wave-particle duality applied not only to photons but to electrons and every other
physical system.

The situation changed rapidly in the years 1925—1930, when working mathematical foundations were found through
the groundbreaking work of Erwin Schrodinger, Werner Heisenberg, Max Born, Pascual Jordan, and the
foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several
different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in
these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of
complementarity .

The "new quantum theory"

Erwin Schrodinger's wave mechanics originally was the first successful attempt at replicating the observed
quantization of atomic spectra with the help of a precise mathematical realization of de Broglie's wave-particle
duality. Schrodinger's wave mechanics were created independently, uniquely based on de Broglie's concepts, less
formal and easier to understand, visualize and exploit. Within a year, it was shown that the two theories were
equivalent. Schrodinger himself initially did not understand the fundamental probabilistic nature of quantum
mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the
charge density of an object smeared out over an extended, possibly infinite, volume of space, but Max Born
introduced the interpretation of the absolute square of the wave function as the probability distribution of the position
of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen, who then became the "father" of
the Copenhagen interpretation of quantum mechanics. Schrodinger's wave function can be seen to be closely related
to the classical Hamilton— Jacobi equation. The correspondence to classical mechanics was even more explicit,
although somewhat more formal, in Heisenberg's matrix mechanics. I.e., the equation for the operators in the
Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain
quantities in the Hamiltonian formalism of classical mechanics, where one uses Poisson brackets.

To be more precise: already before Schrodinger the young student Werner Heisenberg invented his matrix
mechanics, which was the first correct quantum mechanics, i.e. the essential breakthrough. Heisenberg's matrix
mechanics formulation was based on algebras of infinite matrices, being certainly very radical in light of the
mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time,
not even knowing that his "index-schemes" were matrices. In fact, in these early years linear algebra was not
generally known to physicists in its present form.

Although Schrodinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix
mechanics, the reconciliation of the two approaches is generally attributed to Paul Dirac, who wrote a lucid account
in his 1930 classic Principles of Quantum Mechanics, being the third, and perhaps most important, person working
independently in that field (he soon was the only one, who found a relativistic generalization of the theory). In his

Mathematical Formulations of Quantum Dynamics

above-mentioned account, he introduced the bra-ket notation, together with an abstract formulation in terms of the
Hilbert space used in functional analysis; he showed that Schrodinger's and Heisenberg's approaches were two
different representations of the same theory and found a third, most general one, which represented the dynamics of
the system. His work was particularly fruitful in all kind of generalizations of the field. Concerning quantum
mechanics, Dirac's method is now called canonical quantization.

The first complete mathematical formulation of this approach is generally credited to John von Neumann's 1932
book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert
spaces (which he called unitary spaces) in his 1927 classic book. It was developed in parallel with a new approach to
the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's
approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic
framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be
traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the
theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical
foundations.

Later developments

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was
developed starting around 1930. Quantum field theory has driven the development of more sophisticated
formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties
involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory
fashion except for ordinary quantum mechanics.

• Feynman path integrals

• axiomatic, algebraic and constructive quantum field theory

• geometric quantization

• quantum field theory in curved spacetime

• C* algebra formalism

• Generalized Statistical Model of Quantum Mechanics

On a different front, von Neumann originally dispatched quantum measurement with his infamous postulate on the
collapse of the wavefunction, raising a host of philosophical problems. Over the intervening 70 years, the problem of
measurement became an active research area and itself spawned some new formulations of quantum mechanics.

• Relative state/Many-worlds interpretation of quantum mechanics

• Decoherence

• Consistent histories formulation of quantum mechanics

• Quantum logic formulation of quantum mechanics

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to
successful old theories in some approximation. For quantum mechanics, this translates into the need to study the
so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language
are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than
quantum ones. In particular, quantization, namely the construction of a quantum theory whose classical limit is a
given and known classical theory, becomes an important area of quantum physics in itself.

Finally, some of the originators of quantum theory (notably Einstein and Schrodinger) were unhappy with what they
thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that
quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue
of hidden variables has become in part an experimental issue with the help of quantum optics.

• de Broglie— Bohm— Bell pilot wave formulation of quantum mechanics

• Bell's inequalities

Mathematical Formulations of Quantum Dynamics

• Kochen— Specker theorem

Mathematical structure of quantum mechanics

A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of
time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly
direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are
real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the
phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of
a Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a
one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized
by unitary transformations.

Postulates of quantum mechanics

The following summary of the mathematical framework of quantum mechanics can be partly traced back to von
Neumann's postulates.

• Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product
{^IV") ■ Rays (one-dimensional subspaces) in H are associated with states of the system. In other words, physical
states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same
state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the
physical interpretation that countably many observations are enough to uniquely determine the state.

• The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the
component systems (for instance, J.M. Jauch, Foundations of quantum mechanics, section 11-7). For a
non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the
individual particles.

• Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem
(supersymmetry is another matter entirely).

• Physical observables are represented by densely-defined self-adjoint operators on H.

The expected value (in the sense of probability theory) of the observable A for the system in state represented
by the unit vector |V>}e His

(tP\A\^)
By spectral theory, we can associate a probability measure to the values of A in any state i|). We can also show
that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A
has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.

More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative
self-adjoint operator pnormalized to be of trace 1. The expected value of A in the state pis

ti(Ap)

If Pip is the orthogonal projector onto the one-dimensional subspace of H spanned by |V>) , then

ti(Ap^) = (ip | A I ip)

Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal
projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density
operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators
mixed states.

One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact
historical sequence of events, concerning who derived what and under which framework, is the subject of historical

Mathematical Formulations of Quantum Dynamics 90

investigations outside the scope of this article.

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin
and Pauli's exclusion principle, see below.

Superselection sectors. The correspondence between states and rays needs to be refined somewhat to take into
account so-called superselection sectors. States in different superselection sectors cannot influence each other, and
the relative phases between them are unobservable.

Pictures of dynamics

• In the so-called Schrodinger picture of quantum mechanics, the dynamics is given as follows:

The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of
time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If
1-0 (£)\ denotes the state of the system at any one time t, the following Schrodinger equation holds:

ihj t m)) = H\m)

where H is a densely-defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and /j, is
the reduced Planck constant. As an observable, H corresponds to the total energy of the system.

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U(t):
H — > H such that

m + s)) = U(t)\i;(s))
for all times s, t. The existence of a self-adjoint Hamiltonian H such that

U{t) = e~W tH
is a consequence of Stone's theorem on one-parameter unitary groups. (It is assumed that H does not depend on time
and that the perturbation starts at £ = 0; otherwise one must use the Dyson series, formally written as

U{t) = T {exp -(i/h) f dt'H(t')} ,

to
where fis Dyson's time-ordering symbol.

(This symbol permutes a product of noncommuting operators of the form

Bi(ti) ■ B 2 (t 2 ) ■...■ B n {t n )

into the uniquely determined re-ordered expression

B n {t n ) ■ B l2 {U 2 ) ■...■ S^Jwith t n > U a > ■ ■ ■ > ^ ■

The result is a causal chain, the primary cause in the past on the utmost r.h.s., and finally the present effect on the
utmost l.h.s. .)

• The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as
varying in time, it regards the states as fixed and the observables as changing. To go from the Schrodinger to the
Heisenberg picture one needs to define time-independent states and time-dependent operators thus:

|V) = |V(0)>

A(t) = U(-t)AU(t).

It is then easily checked that the expected values of all observables are the same in both pictures

^ i A(t) i if,) = <v>(*) i a i m)

and that the time-dependent Heisenberg operators satisfy

ih±A(t) = [A(t),K\.

Mathematical Formulations of Quantum Dynamics 91

This assumes A is not time dependent in the Schrodinger picture. Notice the commutator expression is purely formal
when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.

• The so-called Dirac picture or interaction picture has time-dependent states and observables, evolving with

respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved
exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the
observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In
symbols:

ihj t A(t) = [A(t),H ].

The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states
that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an
interacting part within a superselection sector. Moreover, even if in the Schrodinger picture the Hamiltonian does not
depend on time, e.g. H = Ho + V> m me interaction picture it does, at least, if V does not commute with Ho,
since

Hint (t) = e (Wtflo Ve H/s)«Jc.
So the above-mentioned Dyson-series has to be used anyhow.

The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in
the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed",
and the Schrodinger picture is considered easiest to visualize and understand by most people, to judge from
pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is
specially associated to quantum field theory and many-body physics.

Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time
would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a
translation distance) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry
(for instance, angular or linear momentum).

Representations

The original form of the Schrodinger equation depends on choosing a particular representation of Heisenberg's
canonical commutation relations. The Stone— von Neumann theorem states all irreducible representations of the
finite-dimensional Heisenberg commutation relations are unitarily equivalent. This is related to quantization and the
correspondence between classical and quantum mechanics, and is therefore not strictly part of the general
mathematical framework.

The quantum harmonic oscillator is an exactly-solvable system where the possibility of choosing among more than
one representation can be seen in all its glory. There, apart from the Schrodinger (position or momentum)
representation one encounters the Fock (number) representation and the Bargmann-Segal (phase space or coherent
state) representation. All three are unitarily equivalent.

Time as an operator

The framework presented so far singles out time as the parameter that everything depends on. It is possible to
formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator. At
the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical
parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the
quantum level, translations in s would be generated by a "Hamiltonian" H - E, where E is the energy operator and H

Mathematical Formulations of Quantum Dynamics 92

is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by
"s-evolution", and so the physical state space is the kernel of H - E (this requires the use of a rigged Hilbert space
and a renormalization of the norm).

This is related to quantization of constrained systems and quantization of gauge theories. It is also possible to
formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).

Spin

In addition to their other properties all particles possess a quantity, which has no correspondence at all in
conventional physics, namely the spin, which is some kind of intrinsic angular momentum (therefore the name). In
the position representation, instead of a wavefunction without spin, ip = tp(r), one has with spin: tp = ip(r, <r)
, where u belongs to the following discrete set of values

a G {-S -h,-(S-l)-h,..., +(S -l)-h,+S-h}.

One distinguishes bosons (S = or 1 or 2 or ...) and fermions (S = 111 or 3/2 or 5/2 or ...)

Pauli's principle

The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion
principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in
the position representation one must postulate that for the transposition of any two of the N particles one always
should have

i/>( . . . ; r i5 (n\ . . . ; r J5 a j ; . . . ) = (-l) 2S ■ i/,( . . . ; r i: a 5 ; . . . ; r i; a { ; . . . )

i.e., on transposition of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor

25
(-1) which is +1 for bosons, but (-1) for fermions. Electrons are fermions with S = 1/2; quanta of light are bosons

with S= 1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum

theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic

part. Only in dimension d=2 one can construct entities where f_l) 2S is replaced by an arbitrary complex number

with magnitude 1 ( -> anyons).

Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics the
properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit.
Especially, many important properties in natural science, e.g. the periodic system of chemistry, are consequences of
the two properties.

The problem of measurement

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However,

it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the

T21
effects of measurement. The von Neumann description of quantum measurement of an observable A, when the

system is prepared in a pure state \p is the following (note, however, that von Neumann's description dates back to

the 1930s and is based on experiments as performed during that time — more specifically the Compton— Simon

experiment; it is not applicable to most present-day measurements within the quantum domain):

• Let A have spectral resolution

A = J\dE A (\),

where E is the resolution of the identity (also called projection-valued measure) associated to A. Then the

probability of the measurement outcome lying in an interval B of R is IE (B) ip\ . In other words, the probability is
obtained by integrating the characteristic function of B against the countably additive measure

Mathematical Formulations of Quantum Dynamics 93

<V> I e a v}-

• If the measured value is contained in B, then immediately after the measurement, the system will be in the
(generally non-normalized) state E (B) \p. If the measured value does not lie in B, replace B by its complement
for the above state.

For example, suppose the state space is the n-dimensional complex Hilbert space C" and A is a Hermitian matrix
with eigenvalues A, with corresponding eigenvectors \p .. The projection-valued measure associated with A, E , is
then

where B is a Borel set containing only the single eigenvalue X.. If the system is prepared in state

Then the probability of a measurement returning the value X . can be calculated by integrating the spectral measure

{1> I E^V)

over B .. This gives trivially

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will
give the same results. This is also called the projection postulate.

A more general formulation replaces the projection-valued measure with a positive-operator valued measure
(POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

by a finite set of positive operators

FF*

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes
{X ... X } is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement
outcome is X .. Instead of collapsing to the (unnormalized) state

after the measurement, the system now will be in the state

Since the F. F* 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer
holds.

The same formulation applies to general mixed states.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution
in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic
and non-unitary. However, since both types of state transformation take one quantum state to another, this difference
was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other
quantum operations, which are described by completely positive maps which do not increase the trace.

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only
the quantum system, but also, and essentially, the classical measurement apparatus (see above).

Mathematical Formulations of Quantum Dynamics 94

The relative state interpretation

An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed the
"many-worlds interpretation" of quantum mechanics.

List of mathematical tools

Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put
together by Richard Courant from David Hilbert's Gottingen University courses. The story is told (by
mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the
advent of Schrodinger's equation. At that point it was realised that the mathematics of the new quantum mechanics
was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and
Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations,
advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years
later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the
physics was radically new.

The main tools include:

• linear algebra: complex numbers, eigenvectors, eigenvalues

• functional analysis: Hilbert spaces, linear operators, spectral theory

• differential equations: partial differential equations, separation of variables, ordinary differential equations,
Sturm— Liouville theory, eigenfunctions

• harmonic analysis: Fourier transforms

See also: list of mathematical topics in quantum theory.

References

• T.S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894—1912, Clarendon Press, Oxford and Oxford
University Press, New York, 1978.

• S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995.

• D. Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, 42 (1979),pp. 1—70.

• G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972.

• J.M. Jauch, Foundations of quantum mechanics, Addison-Wesley Publ. Cy., Reading, Mass., 1968.

• R. Jost, The General Theory of Quantized Fields, American Mathematical Society, 1965.

• A. Gleason, Measures on the Closed Subspaces of a Hilbert Space, Journal of Mathematics and Mechanics, 1957.

• G Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by
Dover 2004).

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. Reprinted
in paperback form.

• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin 1964 (Reprinted by Princeton
University Press)

• M. Reed and B. Simon, Methods of Mathematical Physics, vols I— IV, Academic Press 1972.

• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators, http://
www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.

• N. Weaver, "Mathematical Quantization", Chapman & Hall/CRC 2001.

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.

Mathematical Formulations of Quantum Dynamics 95

Notes

[1] http://books.google.com/books?id=D2Xs8NUKecAC&pg=PA277&lpg=PA277&dq=mathematical+formulation+of+quantum+

mechanics&source=bl&ots=hV5VX7FfDj&sig=JuVbSojzBrKJ9MLKrEGGvqLn9SE&hl=en&ei=BTsvSq6CL6amM8PVhIcK&sa=X&

oi=book_result&ct=result&resnum=8
[2] http://books.google.com/books?id=5t0tm0FBlCsC&pg=PA215&lpg=PA215&dq=wave+function+collapse&source=bl&

ots=a7iUGurRDC&sig=olddjY71Qrj4EQdvS49xcceWq2M&hl=en&ei=RfgtSsDNL4WgM8u-rf4J&sa=X&oi=book_result&ct=result&

resnum=7#PPA215,Ml

Quantum Chemistry and Biochemical Dynamics

Quantum chemistry applies quantum theory to the explanation and prediction of chemical behaviour. The majority
of quantum chemical investigations, at the time of writing, determine the energies of molecules, using heavy
computations based on approximate solutions of the Schrodinger equation. Other quantum chemical studies use
semi-empirical and other methods that are also based on quantum mechanical principles, and deal with time
dependent problems. Quantum chemical studies relate to the ground state of individual atoms and molecules, to
excited states, and to the transition states that occur during chemical reactions. Quantum chemical results include
molecular geometry, the strengths and other characteristics of chemical bonds, optical and other spectra,
intermolecular forces, chemical reactivity and many other chemical properties and features of chemical behaviour.
Many quantum chemical studies assume the nuclei are at rest (Born-Oppenheimer approximation). Many
calculations involve iterative methods that include self-consistent field methods. Major goals of quantum chemistry
include increasing the accuracy of the results for small molecular systems, and increasing the size of large molecules
that can be processed, which is limited by scaling considerations — the computation time increases as a power of the
number of atoms.

An alternative approach

Quantum chemistry is a branch of theoretical chemistry which applies quantum mechanics and quantum field
theory to address problems in chemistry. One application of quantum chemistry is the electronic behavior of atoms
and molecules relative to their chemical reactivity. Quantum chemistry lies on the border between chemistry and
physics. Thus, significant contributions have been made by scientists from both fields. It has a strong and active
overlap with the field of atomic physics and molecular physics, as well as physical chemistry.

Quantum chemistry mathematically describes the fundamental behavior of matter at the molecular scale, but can

span from elementary particles such as electrons (fermions) and photons (bosons) to the cosmos such as star

T21
formation. It is, in principle, possible to describe all chemical systems using this theory. In practice, only the

simplest chemical systems may realistically be investigated in purely quantum mechanical terms, and

approximations must be made for most practical purposes (e.g., Hartree-Fock, post Hartree-Fock or Density

functional theory, see computational chemistry for more details). Hence a detailed understanding of quantum

mechanics is not necessary for most chemistry, as the important implications of the theory (principally the orbital

approximation) can be understood and applied in simpler terms.

In quantum mechanics the Hamiltonian, or the physical state, of a particle can be expressed as the sum of two
operators, one corresponding to kinetic energy and the other to potential energy. The Hamiltonian in the Schrodinger
wave equation used in quantum chemistry does not contain terms for the spin of the electron.

Solutions of the Schrodinger equation for the hydrogen atom gives the form of the wave function for atomic orbitals,
and the relative energy of the various orbitals. The orbital approximation can be used to understand the other atoms
e.g. helium, lithium and carbon.

Quantum Chemistry and Biochemical Dynamics

History

The history of quantum chemistry essentially began with the 1838 discovery of cathode rays by Michael Faraday, the
1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann
that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that
any energy radiating atomic system can theoretically be divided into a number of discrete energy elements e such
that each of these energy elements is proportional to the frequency v with which they each individually radiate
energy, as defined by the following formula:

e = hv

where h is a numerical value called Planck's Constant. Then, in 1905, to explain the photoelectric effect (1839), i.e.,
that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated,
based on Planck's quantum hypothesis, that light itself consists of individual quantum particles, which later came to
be called photons (1926). In the years to follow, this theoretical basis slowly began to be applied to chemical
structure, reactivity, and bonding.

Electronic structure

The first step in solving a quantum chemical problem is usually solving the Schrodinger equation (or Dirac equation
in relativistic quantum chemistry) with the electronic molecular Hamiltonian. This is called determining the
electronic structure of the molecule. It can be said that the electronic structure of a molecule or crystal implies
essentially its chemical properties. An exact solution for the Schrodinger equation can only be obtained for the
hydrogen atom. Since all other atomic, or molecular systems, involve the motions of three or more "particles", their
Schrodinger equations cannot be solved exactly and so approximate solutions must be sought.

Wave model

The foundation of quantum mechanics and quantum chemistry is the wave model, in which the atom is a small,
dense, positively charged nucleus surrounded by electrons. Unlike the earlier Bohr model of the atom, however, the
wave model describes electrons as "clouds" moving in orbitals, and their positions are represented by probability
distributions rather than discrete points. The strength of this model lies in its predictive power. Specifically, it
predicts the pattern of chemically similar elements found in the periodic table. The wave model is so named because
electrons exhibit properties (such as interference) traditionally associated with waves. See wave-particle duality.

Valence bond

Although the mathematical basis of quantum chemistry had been laid by Schrodinger in 1926, it is generally
accepted that the first true calculation in quantum chemistry was that of the German physicists Walter Heitler and
Fritz London on the hydrogen (H ) molecule in 1927. Heitler and London's method was extended by the American
theoretical physicist John C. Slater and the American theoretical chemist Linus Pauling to become the
Valence-Bond (VB) [or Heitler-London-Slater-Pauling (HLSP)] method. In this method, attention is primarily
devoted to the pairwise interactions between atoms, and this method therefore correlates closely with classical
chemists' drawings of bonds.

Molecular orbital

An alternative approach was developed in 1929 by Friedrich Hund and Robert S. Mulliken, in which electrons are
described by mathematical functions delocalized over an entire molecule. The Hund-Mulliken approach or
molecular orbital (MO) method is less intuitive to chemists, but has turned out capable of predicting spectroscopic
properties better than the VB method. This approach is the conceptional basis of the Hartree-Fock method and
further post Hartree-Fock methods.

Quantum Chemistry and Biochemical Dynamics 97

Density functional theory

The Thomas-Fermi model was developed independently by Thomas and Fermi in 1927. This was the first attempt
to describe many-electron systems on the basis of electronic density instead of wave functions, although it was not
very successful in the treatment of entire molecules. The method did provide the basis for what is now known as
density functional theory. Though this method is less developed than post Hartree-Fock methods, its significantly
lower computational requirements (scaling typically no worse than j^ 3 with respect to n basis functions) allow it to
tackle larger polyatomic molecules and even macromolecules. This computational affordability and often
comparable accuracy to MP2 and CCSD (post-Hartree— Fock methods) has made it one of the most popular methods
in computational chemistry at present.

Chemical dynamics

A further step can consist of solving the Schrodinger equation with the total molecular Hamiltonian in order to study
the motion of molecules. Direct solution of the Schrodinger equation is called quantum molecular dynamics, within
the semiclassical approximation semiclassical molecular dynamics, and within the classical mechanics framework
molecular dynamics (MD). Statistical approaches, using for example Monte Carlo methods, are also possible.

Adiabatic chemical dynamics

In adiabatic dynamics, interatomic interactions are represented by single scalar potentials called potential energy
surfaces. This is the Born-Oppenheimer approximation introduced by Born and Oppenheimer in 1927. Pioneering
applications of this in chemistry were performed by Rice and Ramsperger in 1927 and Kassel in 1928, and
generalized into the RRKM theory in 1952 by Marcus who took the transition state theory developed by Eyring in
1935 into account. These methods enable simple estimates of unimolecular reaction rates from a few characteristics
of the potential surface.

Non-adiabatic chemical dynamics

Non-adiabatic dynamics consists of taking the interaction between several coupled potential energy surface
(corresponding to different electronic quantum states of the molecule). The coupling terms are called vibronic
couplings. The pioneering work in this field was done by Stueckelberg, Landau, and Zener in the 1930s, in their
work on what is now known as the Landau-Zener transition. Their formula allows the transition probability between
two diabatic potential curves in the neighborhood of an avoided crossing to be calculated.

Quantum chemistry and quantum field theory

The application of quantum field theory (QFT) to chemical systems and theories has become increasingly common
in the modern physical sciences. One of the first and most fundamentally explicit appearances of this is seen in the
theory of the photomagneton. In this system, plasmas, which are ubiquitous in both physics and chemistry, are
studied in order to determine the basic quantization of the underlying bosonic field. However, quantum field theory
is of interest in many fields of chemistry, including: nuclear chemistry, astrochemistry, sonochemistry, and quantum
hydrodynamics. Field theoretic methods have also been critical in developing the ab initio Effective Hamiltonian
theory of semi-empirical pi-electron methods.

Quantum Chemistry and Biochemical Dynamics

Further reading

• Atkins, P.W.; Friedman, R. (2005). Molecular Quantum Mechanics (4th ed.). Oxford University Press.
ISBN 978-0-19-927498-7.

• Atkins, P.W.. Physical Chemistry. Oxford University Press. ISBN 0-19-879285-9.

• Atkins, P.W.; Friedman, R. (2008). Quanta, Matter and Change: A Molecular Approach to Physical Change.
ISBN 978-0-7167-6117-4.

• Pullman, Bernard; Pullman, Alberte (1963). Quantum Biochemistry. New York and London: Academic Press.
ISBN 90277 1830X.

• The Periodic Table: Its Story and Its Significance. Oxford University Press. 2006. ISBN 0-19-530573-6.
Considers the extent to which chemistry and especially the periodic system has been reduced to quantum
mechanics.

• McWeeny, R.. Coulson's Valence. Oxford Science Publications. ISBN 0-19-855144-4.

• Karplus M., Porter R.N. (1971). Atoms and Molecules. An introduction for students of physical chemistry ,
Benjamin-Cummings Publishing Company, ISBN 978-0-8053-5218-4

• Landau, L.D.; Lifshitz, E.M.. Quantum Mechanics:Non-relativistic Theory. Course of Theoretical Physic. 3.
Pergamon Press. ISBN 008019012X.

• Levine, I. (2008). Physical Chemistry (6th ed.). McGraw-Hill Science. ISBN 978-0-07-253862-5.

• Pauling, L. (1954). General Chemistry. Dover Publications. ISBN 0-486-65622-5.

• Pauling, L.; Wilson, E. B. (1935/1963). Introduction to Quantum Mechanics with Applications to Chemistry.
Dover Publications. ISBN 0-486-64871-0.

• Simon, Z. (1976). Quantum Biochemistry and Specific Interactions. Taylor & Francis. ISBN 978-0-85626-087-2.

References

[1] "Quantum Chemistry" (http://web.archive.Org/web/20070310201939/http://cmm.cit.nih.gov/modeling/guide_documents/

quantum_mechanics_document.html). The NIH Guide to Molecular Modeling. National Institutes of Health. Archived from the original
(http://cmm.cit.nih.gov/modeling/guide_documents/quantum_mechanics_document.html) on 2007-03-10. . Retrieved 2007-09-08.

[2] "Astrophysics and Astrochemistry". Astrophysics and Astrochemistry . (http://www.astrochemistry.eu/).

External links

• The Sherrill Group - Notes (http://vergil.chemistry.gatech.edu/notes/index.html)

• ChemViz Curriculum Support Resources (http://www.shodor.org/chemviz/)

• Early ideas in the history of quantum chemistry (http://www.quantum-chemistry-history.com/)

• The Particle Adventure (http://particleadventure.org/)

Nobel lectures by quantum chemists

• Walter Kohn's Nobel lecture (http://nobelprize.org/chemistry/laureates/1998/kohn-lecture.html)

• Rudolph Marcus' Nobel lecture (http://nobelprize.org/chemistry/laureates/1992/marcus-lecture.html)

• Robert Mulliken's Nobel lecture (http://nobelprize.org/chemistry/laureates/1966/mulliken-lecture.html)

• Linus Pauling's Nobel lecture (http://nobelprize.org/chemistry/laureates/1954/pauling-lecture.html)

• John Pople's Nobel lecture (http://nobelprize.org/chemistry/laureates/1998/pople-lecture.html)

List of quantum chemistry and solid state physics software

List of quantum chemistry and solid state physics
software

Quantum chemistry computer programs are used in computational chemistry to implement the methods of
quantum chemistry. Most include the Hartree— Fock (HF) and some post-Hartree— Fock methods. They may also
include density functional theory (DFT), molecular mechanics or semi-empirical quantum chemistry methods. The
programs include both open source and commercial software. Most of them are large, often containing several
separate programs, and have developed over many years.

The following table illustrates the capabilities of the most versatile software packages that show an entry in two or
more columns of the table.

Package

License

Lang.

Basis

Periodic

Mol.
mech.

Semi-emp.

Post-HF

DFT

ABINIT

GPL

Fortran

Yes

ACES II

Academic

Fortran

GTO

Yes

ACES II MAB

Academic

Fortran

GTO

Yes

ADF

Commercial

Fortran

STO

Any

Yes

Yes 4

Yes

Atomistix

ToolKit

(ATK)

Commercial

C++/Python

NAO/EHT

3d 9

Yes

BigDFT

GPL

Fortran

Wavelet

Any

Yes

CADPAC

Academic

Fortran

GTO

Yes

CASINO
(QMC)

Academic

Fortran

GTO / PW /
Spline / Grid /
STO

Any

Yes

CASTEP

Academic

(UK)/

Commercial

Fortran

Yes

Yes 5

Yes

CFOUR

Academic

Fortran

GTO

Yes

COLUMBUS

Academic

Fortran

GTO

Yes

CONQUEST

Academic
(UK)

Unknown

COSMOS

Commercial

Unknown

Yes

CP2K

GPL

Fortran 95

Hybrid GTO /
PW

Any

Yes

CPMD

Academic

Fortran

Any

Yes

CRYSTAL

Academic

(UK)/

Commercial

Fortran

GTO

Any

Yes

Yes 10

Yes

DACAPO

GPL? 1

Fortran

Yes

DALTON

Academic

Fortran

GTO

Yes

DFTB+ [1]

Academic /
Commercial

Fortran 95

NAO

Any

Yes

List of quantum chemistry and solid state physics software

100

DFT++ [2]

GPL

C++

PW / Wavelet

Yes

DIRAC

Academic

Fortran 77,
Fortran 90,
C

GTO

Yes

DMol3 [3]

Commercial

Unknown

Numeric AOs

Yes

ErgoSCF [4]

GPL

C++

GTO

Yes

EXCITING

GPL

Fortran 95

FP-LAPW

Unknown

Yes

Unknown

Yes

FLEUR [5]

Academic

Unknown

FP-(L)APW+lo

3d, 2d,

Yes

FHI-aims [6]

Commercial

Fortran

NAO

Any

Yes

FreeON

GPL

Fortran 95

GTO

Any

Yes

Firefly / PC
GAMES S

Academic

Unknown

GTO

Yes 3

Yes

GAMES S

(UK)

Academic

(UK)/

Commercial

Fortran

GTO

Yes

GAMES S

(US)

Academic

Fortran

GTO

Yes 2

Yes

GAUSSIAN

Commercial

Fortran

GTO

Any

Yes

GPAW [7]

GPL

Python / C

Grid / NAO

Any

Yes

Unknown

Yes 5

Yes

hBar Lab

Commercial

Unknown

GTO

Yes

HiLAPW [8]

Unknown

FLAPW

Yes

IAGUAR

Commercial

Unknown

GTO

Unknown

Yes

MADNESS

GPL

C++

Wavelet

Unknown

Yes

MOLCAS

Commercial

Fortran

GTO

Yes

MOLPRO

Commercial

Unknown

GTO

Unknown

Yes

MOPAC

Academic /
Commercial

Fortran

Unknown

Yes

MPQC

LGPL

C++

GTO

Yes

NWChem

Academic

Fortran 77 /
C

Unknown

Yes

OCTOPUS

GPL

Unknown

Grid

Any

Yes

ONETEP

Academic

(UK)/

Commercial

Fortran

Any

Yes

Yes 5

Yes

OpenAtom

Academic

Charm++
(C++)

DVR

Unknown

Yes

OpenMX [9]

GPL

NAO

Yes

ORCA [10]

Academic

C++

GTO

Any

Yes

List of quantum chemistry and solid state physics software

101

PLATO

Academic

Unknown

NAO

Any

Yes

PQS

Commercial

Unknown

Yes

Priroda-06 [11]

Academic

Unknown

GTO

Yes

PSI

GPL

C/C++

GTO

Yes

PWscf 6

GPL

Fortran

Yes

PyQuante

BSD

Python

GTO

Yes

Q-Chem

Commercial

Fortran /
C++

GTO

Yes

Quantemol-N

Academic /
Commercial

Fortran

GTO

Yes

Quantum

ESPRESSO

[12]

GPL

Fortran

Yes

RSPt [13]

Academic

Fortran / C

FP-LMTO

Yes

SPARTAN

Commercial

Unknown

GTO

Yes

SIESTA

Academic

Fortran

NAO

Yes

TB-LMTO

[14]

Academic

Fortran

LMTO

Yes

TERACHEM 8

Commercial

C/CUDA

GTO

Yes

TURBOMOLE

Commercial

Fortran

GTO

Yes

VASP

Academic(AT)/
Commercial

Fortran

Any

Yes

WIEN2k

Commercial

Unknown

FP-(L)APW+lo

Yes

"Academic": academic (no cost) license possible upon request; "Commercial": commercially distributed.

Support for periodic systems (3d-crystals, 2d-slabs, Id-rods and isolated molecules): 3d-periodic codes always
allow the simulation of systems with lower dimensionality within a supercell. Specified here is the capability for
actual simulation within lower periodicity.

The CAMPOS project (which includes Dacapo) states that all code is GPL. The Dacapo distribution itself does
not contain any license information.

2 Through interface to TINKER

Through Ascalaph
4 Through interface to MOPAC

Using exact exchange DFT
6 Distributed with Quantum ESPRESSO [12]

Web service integrating MPQC.

TeraChem is the first fully GPU-accelerated quantum chemistry software.

9
Atomistix ToolKit also contains finite-bias NEGF electron transport calculations with open boundary conditions.

10 Through CRYSCOR [16] program.

List of quantum chemistry and solid state physics software

102

Further programs

AIMPRO L J

Ascalaph Designer

PWPAW / Atompaw

deMon2K [2 ]
[24]

DFTB+ '

Fireball

[26]

FSatom

[18]

MAPS

[19]

NRLMOL

ORCA [1 ° ]
ParaGauss

[20]

[25]

PARATEC
PARSEC

,[21]

Petot

Socorro

[23]

S/PHI/nX

[48]

SPR-KKR

[27]

References

• Young, David (2001). Computational Chemistry: A Practical Guide for Applying Techniques to Real World
Problems. New York: John Wiley & Sons. pp. 322-359. ISBN 0-471-33368-9.

References

[I] http://www.dftb-plus.info

[2] http://dft.physics.cornell.edu/

[3] http://people.web.psi.ch/delley/dmol3.html

[4] http://www.ergoscf.org

[5] http://www.flapw.de

[6] http://www.fhi-berlin.mpg.de/aims/

[7] https://wiki.fysik.dtu.dk/gpaw/

[8] http://home.hiroshima-u.ac.jp/fpc/manuals/HiLAPW/HiLAPW.html

[9] http://www.openmx-square.org/

[10] http://www.thch.uni-bonn.de/tc/orca/

[II] http ://www. phy sto. se/~laikov/p/
[12] http ://www. quantum-espresso. org
[13] http://www.rspt.net

[ 14] http ://www. fkf . mpg.de/andersen/

[15] http://www.camd.dtu.dk/software.aspx

[16] http://www.cryscor.unito.it

[17] http://aimpro.ncl.ac.uk/

[18] http://www.tddft.org/fsatom

[19] http://www.scienomics.com/Products/maps/

[20] http ://quantum.utep. edu/nrlmol/nrlmol. html

[2 1 ] http://hpcrd.lbl. gov/~linwang/PEtot/PEtot. html

[22] http://www.demon-software.com/public_html/program.html

[23] http://dft.sandia.gov/Socorro/mainpage.html

[24] http://www.dftb-plus.info/

[25] http://qcl.theochem.tu-muenchen.de/ParaGauss.html

[26] http://www.fireball-dft.org

[27] http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR

103

Basic Concepts in Symbolic Dynamics

Sequential dynamical system

Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems
which generalize many aspects of for example classical cellular automata, and they provide a framework for
studying asynchronous processes over graphs. The analysis of SDSs uses techniques from combinatorics, abstract
algebra, graph theory, dynamical systems and probability theory.

Definition

An SDS is constructed from the following components:

• A finite graph Y with vertex set v[Y] = { 1,2, ... , n}. Depending on the context the graph can be directed or
undirected.

• A state x for each vertex / of Y taken from a finite set K. The system state is the n-tuple x = (x , x , ... , x ), and
x[i] is the tuple consisting of the states associated to the vertices in the 1 -neighborhood of/ in Y (in some fixed
order).

• A v ertex function f. for each vertex /. The vertex function maps the state of vertex / at time t to the vertex state
at time t + 1 based on the states associated to the 1 -neighborhood of i in Y.

• A word w = (w,, w„, ... , w )overv[in.

12 m

It is convenient to introduce the 7-local maps F . constructed from the vertex functions by

Fi(x) = {x 1 ,X2,...,x i -. 1 ,fi{x[i\),zn. 1 ,...,x n ) .
The word w specifies the sequence in which the y-local maps are composed to derive the sequential dynamical
system map F: K" —> K n as

[F Y , w] = F w{m) O i^-i) O ■ ■ ■ O F w{2 ) o F w{1) .
If the update sequence is a permutation one frequently speaks of a permutation SDS to emphasize this point. The
phase space associated to a sequential dynamical system with map F: a —> K n is the finite directed graph with
vertex set K! 1 and directed edges (x, F(x)). The structure of the phase space is governed by the properties of the graph
Y, the vertex functions (/".)., and the update sequence w. A large part of SDS research seeks to infer phase space
properties based on the structure of the system constituents.

Example

Consider the case where Y is the graph with vertex set {1,2,3} and undirected edges {1,2}, {1,3} and {2,3} (a
triangle or 3-circle) with vertex states from K = {0,1 }. For vertex functions use the symmetric, boolean function nor :
K —> K defined by nor(x,y,z) = (l+x)(l+y)(l+z) with boolean arithmetic. Thus, the only case in which the function
nor returns the value 1 is when all the arguments are 0. Pick w = (1,2,3) as update sequence. Starting from the initial
system state (0,0,0) at time t = one computes the state of vertex 1 at time f=l as nor(0,0,0) = 1. The state of vertex
2 at time f=l is nor( 1,0,0) = 0. Note that the state of vertex 1 at time f=l is used immediately. Next one obtains the
state of vertex 3 at time f=l as nor(l,0,0) = 0. This completes the update sequence, and one concludes that the
Nor-SDS map sends the system state (0,0,0) to (1,0,0). The system state (1,0,0) is in turned mapped to (0,1,0) by an
application of the SDS map.

Sequential dynamical system

104

References

• Henning S. Mortveit, Christian M. Reidys (2008). An Introduction to Sequential Dynamical Systems. Springer.
ISBN 0387306544.

Predecessor and Permutation Existence Problems for Sequential Dynamical Systems

[l]

Genetic Sequential Dynamical Systems

[2]

References

[1] http://www.emis.de/joumals/DMTCS/pdfpapers/dmAB0106.pdf
[2] http://arxiv.org/pdf/math.DS/0603370

Automata theory

In theoretical computer science, automata theory is

the study of abstract machines (or more appropriately,
abstract 'mathematical' machines or systems) and the
computational problems that can be solved using these
machines. These abstract machines are called automata.

The figure at right illustrates a finite state machine,
which is one well-known variety of automaton. This
automaton consists of states (represented in the figure
by circles), and transitions (represented by arrows). As
the automaton sees a symbol of input, it makes a
transition (or jump) to another state, according to its
transition function (which takes the current state and
the recent symbol as its inputs).

Automata theory is also closely related to formal language theory, as the automata are often classified by the class of
formal languages they are able to recognize. An automaton can be a finite representation of a formal language that
may be an infinite set.

Automata play a major role in compiler design and parsing.

An example of automata and study of mathematical properties of
such automata is automata theory

Automata

Following is an introductory definition of one type of automata, which attempts to help one grasp the essential
concepts involved in automata theory.

Informal description

An automaton is supposed to run on some given sequence or string of inputs in discrete time steps. At each time
step, an automaton gets one input that is picked up from a set of symbols or letters, which is called an alphabet. At
any time, the symbols so far fed to the automaton as input form a finite sequence of symbols, which is called a word.
An automaton contains a finite set of states. At each instance in time of some run, automaton is in one of its states.
At each time step when the automaton reads a symbol, it jumps or transits to next state depending on its current state
and on the symbol currently read. This function in terms of the current state and input symbol is called transition
function. The automaton reads the input word one symbol after another in the sequence and transits from state to
state according to the transition function, until the word is read completely. Once the input word has been read, the
automaton is said to have been stopped and the state at which automaton has stopped is called final state. Depending

Automata theory 105

on the final state, it's said that the automaton either accepts or rejects an input word. There is a subset of states of the
automaton, which is defined as the set of accepting states. If the final state is an accepting state, then the automaton
accepts the word. Otherwise, the word is rejected. The set of all the words accepted by an automaton is called the
language recognized by the automaton.

Formal definition

Automaton

An automaton is represented formally by the 5-tuple DQ,S,8,q ,AD, where:

• Q is a finite set of states.

• 2 is a finite set of symbols, called the alphabet of the automaton.

• 5 is the transition function, that is, 6: Q x 2 — » Q.

• q is the start state, that is, the state which the automaton is in when no input has been processed yet, where

q eQ.

• A is a set of states of Q (i.e. ACQ) called accept states.

Input word

An automaton reads a finite string of symbols a ,a ,...., a , where a. G 2, which is called a input word. Set of
all words is denoted by 2*.

Run

A run of the automaton on an input word w = a ,a ,...., a €2*, is a sequence of states q,q,q ,...., q , where

q. € Q such that q„ is the start state and q. = 5(q. ,,a.) for < i < n. In words, at first the automaton is at the
i i i- 1 i

start state q and then automaton reads symbols of the input word in sequence. When automaton reads symbol
a. then it jumps to state q. = 5(q. ,a.). q said to be the, final state of the run.

Accepting word

A word w € 2* is accepted by the automaton if q € A.

Recognized language

An automaton can recognize a formal language. The recognized language L C 2* by an automaton is the set of
all the words that are accepted by the automaton.

Recognizable languages

The recognizable languages is the set of languages that are recognized by some automaton. For above
definition of automata the recognizable languages are regular languages. For different definitions of automata,
the recognizable languages are different.

Variations in definition of automata

Automata are defined to study useful machines under mathematical formalism. So, the definition of an automaton is
open to variations according to the "real world machine", which we want to model using the automaton. People have
studied many variations of automata. Above, the most standard variant is described, which is called deterministic
finite automaton. The following are some popular variations in the definition of different components of automata.

Input

• Finite input: An automaton that accepts only finite sequence of symbols. The above introductory definition only
accepts finite words.

• Infinite input: An automaton that accepts infinite words ( co-words). Such automata are called co-automata.

• Tree word input: The input may be a tree of symbols instead of sequence of symbols. In this case after reading
each symbol, the automaton reads all the successor symbols in the input tree. It is said that the automaton makes
one copy of itself for each successor and each such copy starts running on one of the successor symbol from the

Automata theory 106

state according to the transition relation of the automaton. Such an automaton is called tree automaton.
States

• Finite states: An automaton that contains only a finite number of states. The above introductory definition
describes automata with finite numbers of states.

• Infinite states: An automaton that may not have a finite number of states, or even a countable number of states.
For example, the quantum finite automaton or topological automaton has uncountable infinity of states.

• Stack memory: An automaton may also contain some extra memory in the form of a stack in which symbols can
be pushed and popped. This kind of automaton is called a pushdown automaton

Transition function

• Deterministic: For a given current state and an input symbol, if an automaton can only jump to one and only one
state then it is a deterministic automaton.

• Nondeterministic: An automaton that, after reading an input symbol, may jump into any of a number of states, as
licensed by its transition relation. Notice that the term transition function is replaced by transition relation: The
automaton non-deterministically decides to jump into one of the allowed choices. Such automaton are called
nondeterministic automaton.

• Alternation: This idea is quite similar to tree automaton, but orthogonal. The automaton may run its multiple
copies on the same next read symbol. Such automata are called alternating automaton. Acceptance condition
must satisfy all runs of such copies to accept the input.

Acceptance condition

• Acceptance of finite words: Same as described in the informal definition above.

• Acceptance of infinite words: an omega automaton cannot have final states, as infinite words never terminate.
Rather, acceptance of the word is decided by looking at the infinite sequence of visited states during the run.

• Probabilistic acceptance: An automaton need not strictly accept or reject an input. It may accept the input with
some probability between zero and one. For example, quantum finite automaton, geometric automaton and metric
automaton has probabilistic acceptance.

Different combinations of the above variations produce many variety of automaton.

Automata theory

Automata theory is a subject matter which studies properties of various types of automata. For example, following
questions are studied about a given type of automata.

• Which class of formal languages is recognizable by some type of automata? (Recognizable languages)

• Is certain automata closed under union, intersection, or complementation of formal languages? (Closure
properties)

• How much is a type of automata expressive in terms of recognizing class of formal languages? And, their relative
expressive power? (Language Hierarchy)

Automata theory also studies if there exist any effective algorithm or not to solve problems similar to following list.

• Does an automaton accept any input word? (emptiness checking)

• Is it possible to transform a given non-deterministic automaton into deterministic automaton without changing the
recognizing language? (Determinization)

• For a given formal language, what is the smallest automaton that recognizes it? (Minimization).

Automata theory

107

Classes of automata

Following is an incomplete list of some types of automata.

Automata

Recognizable language

Deterministic finite automata (DFA)

regular languages

Nondeterministic finite automata (NFA)

regular languages

Nondeterministic finite automata with e-transitions (FND-e or e-NFA)

regular languages

Pushdown automata (PDA)

context-free languages

Linear bounded automata (LBA)

context-sensitive language

Turing machines

recursively enumerable languages

Timed automata

Deterministic Buchi automata

(D-limit languages

Nondeterministic Buchi automata

(o-regular languages

Nondeterministic/Deterministic Rabin automata

(o-regular languages

Nondeterministic/Deterministic Streett automata

(D-regular languages

Nondeterministic/Deterministic parity automata

(o-regular languages

Nondeterministic/Deterministic Muller automata

a>-regular languages

Discrete, continuous, and hybrid automata

Normally automata theory describes the states of abstract machines but there are analog automata or continuous
automata or hybrid discrete-continuous automata, using analog data, continuous time, or both.

Applications

Each model in automata theory play varied roles in several applied areas. Finite automata is used in text processing,
compilers, and hardware design. Context-free grammar is used in programming languages and artificial intelligence.
Originally, CFG were used in the study of the human languages. Cellular automata is used in the field of biology, the
most common example being John Conway's Game of Life. Some other examples which could be explained using
automata theory in biology include mollusk and pine cones growth and pigmentation patterns. Going further, a
theory suggesting that the whole universe is computed by some sort of a discrete automaton, is being advocated by
some scientist. The idea originated in the work of Konrad Zuse, most importantly his 1969 book Rechnender Raum
and gave rise to Digital physics.

References

• John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman (2000). Introduction to Automata Theory, Languages, and
Computation (2nd Edition) . Pearson Education. ISBN 0-201-44124-1.

• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Part
One: Automata and Languages, chapters 1—2, pp.29— 122. Section 4.1: Decidable Languages, pp.152— 159.
Section 5.1: Undecidable Problems from Language Theory, pp.172— 183.

• James P. Schmeiser, David T. Barnard (1995). Producing a top-down parse order with bottom-up parsing.
Elsevier North-Holland.

Automata theory

108

External links

• Visual Automata Simulator (http://www.cs.usfca.edu/~jbovet/vas.html), A tool for simulating, visualizing
and transforming finite state automata and Turing Machines, by Jean Bovet
JFLAP (http://www.jflap.org)
dk. brie s . automaton (http :// w w w . brics . dk/ automaton)
libf a (http :// w w w . augeas . net/libf a/index .html)
Proyecto SEPa (in Spanish) (http://www.ucse.edu.ar/fma/sepa/)
Exorciser (in German) (http://www.swisseduc.ch/informatik/exorciser/index.html)
Automata Made it on Java, with the Sources so you can see (http://torturo.com/programas-hechos-en-java/)

Time series analysis

In statistics, signal processing, econometrics and
mathematical finance, a time series is a sequence of
data points, measured typically at successive times
spaced at uniform time intervals. Examples of time
series are the daily closing value of the Dow Jones
index or the annual flow volume of the Nile River at
Aswan. Time series analysis comprises methods for
analyzing time series data in order to extract
meaningful statistics and other characteristics of the
data. Time series forecasting is the use of a model to
forecast future events based on known past events to
predict data points before they are measured. An
example of time series forecasting in econometrics is
predicting the opening price of a stock based on its past
performance. Time series are very frequently plotted
via line charts.

Time series data have a natural temporal ordering. This makes time series analysis distinct from other common data
analysis problems, in which there is no natural ordering of the observations (e.g. explaining people's wages by
reference to their education level, where the individuals' data could be entered in any order). Time series analysis is
also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g.
accounting for house prices by the location as well as the intrinsic characteristics of the houses). A time series model
will generally reflect the fact that observations close together in time will be more closely related than observations
further apart. In addition, time series models will often make use of the natural one-way ordering of time so that
values for a given period will be expressed as deriving in some way from past values, rather than from future values
(see time reversibility.)

Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain
methods. The former include auto-correlation, cross-correlation analysis, spectral analysis and recently wavelet
analysis; auto-correlation and cross-correlation analysis can also be completed in the time domain.

Time series analysis 109

Analysis

There are several types of data analysis available for time series which are appropriate for different purposes.

General exploration

• Graphical examination of data series

• Autocorrelation analysis to examine serial dependence

• Spectral analysis to examine cyclic behaviour which need not be related to seasonality. For example, sun spot
activity varies over 1 1 year cycles. Other common examples include celestial phenomena, weather patterns,
neural activity, commodity prices, and economic activity.

Description

• Separation into components representing trend, seasonality, slow and fast variation, cyclical irregular: see
decomposition of time series

• Simple properties of marginal distributions

Prediction and forecasting

• Fully formed statistical models for stochastic simulation purposes, so as to generate alternative versions of the
time series, representing what might happen over non-specific time-periods in the future

• Simple or fully formed statistical models to describe the likely outcome of the time series in the immediate future,
given knowledge of the most recent outcomes (forecasting).

Models

Models for time series data can have many forms and represent different stochastic processes. When modeling
variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models,
the integrated (I) models, and the moving average (MA) models. These three classes depend linearly on previous
data points. Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive
integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA)
model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under
the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an
initial "V" for "vector". An additional set of extensions of these models is available for use where the observed
time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the
distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's
control. For these models, the acronyms are extended with a final "X" for "exogenous".

Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility
of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage
of using predictions derived from non-linear models, over those from linear models.

Among other types of non-linear time series models, there are models to represent the changes of variance along
time (heteroskedasticity). These models are called autoregressive conditional heteroskedasticity (ARCH) and the
collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc).
Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in
contrast to other possible representations of locally varying variability, where the variability might be modelled as
being driven by a separate time-varying process, as in a doubly stochastic model.

In recent work on model-free analyses, wavelet transform based methods (for example locally stationary wavelets
and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution)
techniques decompose a given time series, attempting to illustrate time dependence at multiple scales.

Time series analysis 110

Notation

A number of different notations are in use for time-series analysis. A common notation specifying a time series X
that is indexed by the natural numbers is written

x={x v x 2 ,...}.

Another common notation is

Y= {Y:te T],
where T is the index set.

Conditions

There are two sets of conditions under which much of the theory is built:

• Stationary process

• Ergodicity

However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and second-order
stationarity. Both models and applications can be developed under each of these conditions, although the models in
the latter case might be considered as only partly specified.

In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary.
Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency
analysis which makes use of a time— frequency representation of a time-series or signal.

Models

The general representation of an autoregressive model, well-known as AR(p), is

Y t = a + ajYt-i + a 2 Y t _ 2 H h a p Y t _ p + e t

where the term e is the source of randomness and is called white noise. It is assumed to have the following
characteristics:

l- E[e t ] =

2. Etf\ = a 2

3. E[e t e s ] = Vt^s

With these assumptions, the process is specified up to second-order moments and, subject to conditions on the
coefficients, may be second-order stationary.

If the noise also has a normal distribution, it is called normal white noise (denoted here by Normal-WN):

i £ t}(teT) ■ Normal-WN.

In this case the AR process may be strictly stationary, again subject to conditions on the coefficients.

Time series analysis 111

Related tools

Tools for investigating time-series data include:

• Consideration of the autocorrelation function and the spectral density function (also cross-correlation functions
and cross-spectral density functions)

• Performing a Fourier transform to investigate the series in the frequency domain

• Use of a filter to remove unwanted noise

• Principal components analysis (or empirical orthogonal function analysis)

• Singular spectrum analysis

• Artificial neural networks

• Time-frequency analysis techniques:

• Fast Fourier Transform

• Continuous wavelet transform

• Short-time Fourier transform

• Chirplet transform

• Fractional Fourier transform

• Chaotic analysis

• Correlation dimension

• Recurrence plots

• Recurrence quantification analysis

• Lyapunov exponents

References

[1] Bloomfield, P. (1976). Fourier analysis of time series: An introduction. New York: Wiley.
[2] Shumway, R. H. (1988). Applied statistical time series analysis. Englewood Cliffs, NJ: Prentice Hall.
[3] Gershenfeld, N. (1999). The nature of mathematical modeling, p. 205-08

[4] Boashash, B. (ed.), (2003) Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003
ISBN ISBN 0080443354

Further reading

Bloomfield, P. (1976). Fourier analysis of time series: An introduction. New York: Wiley.

Box, George; Jenkins, Gwilym (1976), Time series analysis: forecasting and control, rev. ed., Oakland,

California: Holden-Day

Brillinger, D. R. (1975). Time series: Data analysis and theory. New York: Holt, Rinehart. & Winston.

Brigham, E. O. (1974). The fast Fourier transform. Englewood Cliffs, NJ: Prentice-Hall.

Elliott, D. F., & Rao, K. R. (1982). Fast transforms: Algorithms, analyses, applications. New York: Academic

Press.

Gershenfeld, Neil (2000), The nature of mathematical modeling, Cambridge: Cambridge Univ. Press,

ISBN 978-0521570954, OCLC 174825352

Hamilton, James (1994), Time Series Analysis, Princeton: Princeton Univ. Press, ISBN 0-691-04289-6

Jenkins, G M., & Watts, D. G. (1968). Spectral analysis and its applications. San Francisco: Holden-Day.

Priestley, M. B. (1981). Spectral analysis and time series. New York: Academic Press.

Shasha, D. (2004), High Performance Discovery in Time Series, Berlin: Springer, ISBN 0387008578

Shumway, R. H. (1988). Applied statistical time series analysis. Englewood Cliffs, NJ: Prentice Hall.

Wiener, N.(1964). Extrapolation, Interpolation, and Smoothing of Stationary Time Series.The. MIT Press.

Wei, W. W. (1989). Time series analysis: Univariate and multivariate methods. New York: Addison-Wesley.

Time series analysis 112

External links

• A First Course on Time Series Analysis (http://statistik.mathematik.uni-wuerzburg.de/timeseries/) - an open
source book on time series analysis with SAS

• Introduction to Time series Analysis (Engineering Statistics Handbook) (http://www.itl.nist.gov/div898/
handbook/pmc/section4/pmc4.htm) - A practical guide to Time series analysis

• List of Free Software for Time Series Analysis (http://ces.stat.ucla.edu/software/time-series-analysis)

• Online Tutorial 'Recurrence Plot' (Flash animation); lots of examples (http://www.as-internetdienst.de/r67tze4/
einbettung.html)

Lag operator

"Backshift" redirects here. For the linguistic sense see Sequence of tenses.

In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce
the previous element. For example, given some time series

X = {X lt X 2i ...}

then

LX t = Xt-ifor all t > 1
where L is the lag operator. Sometimes the symbol B for backshift is used instead. Note that the lag operator can be
raised to arbitrary integer powers so that

L X t = X t+ i

and

rk x _

L k X t = X,

Lag polynomials

Also polynomials of the lag operator can be used, and this is a common notation for ARMA models. For example,
s t = X t -J2 ViXt-i = ( 1 - £ VilA X t

i=l \ £=1 J

specifies an AR(p) model.

A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely
specified as

<pX t = 9e t
where cp and 6 respectively represent the lag polynomials,
p

V = 1 - £ <PiL %

i=i
and

i=l

An annihilator operator, denoted [ 1 , , removes the entries of the polynomial with negative power (future values).

Lag operator 113

Difference operator

In time series analysis, the first difference operator ^\is a special case of lag polynomial.

AX t =X t -X t _ 1
AX t ={l-L)X t
Similarly, the second difference operator

A(AXt) = AX t - AI M
A 2 X t = (1 - L)AX t
A 2 X t = (1 - L)(l - L)X t
A 2 X t = (1 - LfX t
The above approach generalises to the i 'th difference operator A l Xt = (1 — LYX^

Conditional expectation

It is common in stochastic processes to care about the expected value of a variable given a previous information set.
Let f2fbe all information that is common knowledge at time t (this is often subscripted below the expectation
operator), then the expected value of X that is some j time-steps in the future can be written equivalently as:

E[X t+j \n t ] = E t [X t+j ] .
With these time-dependent conditional expectations, there is the need to distinguish between the Backshift operator
(B) that only adjusts the date of the forecasted variable and the Lag operator (L) that adjusts equally the date of the
forecasted variable and the information set:

L n E t [X t+ j] — E t _ n [X t+ j_ n ] ,
B n E t [X t+j ] = E t [X t+j _ n ] .

References

• Hamilton, James Douglas (1994). Time series analysis. Princeton University Press. ISBN 0-691-04289-6.

• Verbeek, Marno (2008). A guide to modern econometrics. John Wiley and Sons. ISBN 0-470-51769-7.

Shift operator 1 14

Shift operator

In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important
for their simplicity and natural occurrence. They are used in diverse areas, such as Hardy spaces, the theory of
abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation.
(There is another usage of shift operator as a translation operator: see for example Sheffer sequence.) In time series
analysis, this operator is called the lag operator.

A typical one-sided shift operator takes an infinite sequence of numbers

(Q,a v a 2 , ...).
This operation respects typical convergence conditions, such as absolute convergence of the corresponding infinite
series; it therefore gives rise to continuous operators on the standard sequence spaces used in functional analysis,
usually with norm 1 .

Another way to look at it would be in terms of polynomials: the sequences that eventually end in a string

(...,0,0,0,...)

or, in other words, having only a finite number of non-zero entries, are in a 1-1 correspondence with polynomials in
an indeterminate T having a . as coefficient of T 1 . The advantage of this representation is then that the shift operator
becomes multiplication by T: this reveals quickly several aspects of its structure. Spaces of polynomials carry
numerous topological structures; shift operators can be constructed by extension on corresponding complete spaces.

The bilateral shift operators are the related operators in which the sequences are bi-infinite (functions on the
integers, rather than just the natural numbers). One can say that the analogue in this case of the polynomial
representation is that by Laurent polynomials. The theory of analytic functions is related to that of polynomials, by
allowing infinite power series; on the other hand meromorphic functions have Laurent series that terminate in the
direction of negative exponents. In the same way, the one-sided and bilateral shifts have rather different properties.
This connection with function theory is made more precise in the context of Hardy spaces.

Action on Hilbert spaces

The unilateral and bilateral shifts have a natural action on Hilbert spaces, giving bounded operators S and U on the v
sequence spaces £ 2 (N)and ^(^respectively. The unilateral shift S is a proper isometry with range equal to all

vectors which vanish in the first coordinate. The bilateral shift U, on the other hand, is a unitary operator. The
operator S is a compression of U, in the sense that

Ux' = Sx for each x € £ 2 (N),

where ^'is the vector in ^ 2 (^)with x'- = 2^ for i > fjand x'- = Ofor % < 0- This observation is at the heart

of the construction of many unitary dilations of isometries.

The spectrum of S is the unit disk while the spectrum of U is the unit circle in the complex plane.

The Wold decomposition says that every isometry on a Hilbert space is of the form

s a ®u

where S a is S to the power of some cardinal number a and U is a unitary operator. In turn, the C*-algebra generated
by an arbitrary proper isometry is isomorphic to the C*-algebra generated by S.

The shift S is one example of a Fredholm operator; it has Fredholm index - 1 .

Shift operator 115

Bibliography

• Jonathan R. Partington, Linear Operators and Linear Systems, An Analytical Approach to Control Theory, (2004)
London Mathematical Society Student Texts 60, Cambridge University Press.

• Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.

Shift space

In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words
representing the evolution of a discrete system. In fact, shift spaces and symbolic dynamical systems are often
considered synonyms.

Notation

Let A be a finite set of states. An infinite (respectively bi-infinite) word over A is a sequence x = (x TJ )„,gjvf , where
M = PJ (resp. M = Z ) an d x n is m A for any integer n. The shift operator a acts on an infinite or bi-infinite
word by shifting all symbols to the left, i.e.,

(<r(x))(n) = a^+ifor all n.
In the following we choose ][f = pj and thus speak of infinite words, but all definitions are naturally generalizable
to the bi-infinite case.

Definition

A set of infinite words over A is a shift space if it is closed with respect to the natural product topology of j^ and
invariant under the shift operator. Thus a set g C A^ * s a subshift if and only if

1. for any (pointwise) convergent sequence (xfc)fc>o°f elements of S, the limit ^ lrn x fe also belongs to S; and

2. a(S) = S.

A shift space S is sometimes denoted as (S, <r)to emphasize the role of the shift operator.

Some authors use the term subshift for a set of infinite words that is just invariant under the shift, and reserve the

term shift space for those that are also closed.

Characterization and sofic subshifts

A subset S of j^ is a shift space if and only if there exists a set X of finite words such that S coincides with the set
of all infinite words over A having no factor in X,

When X is a regular language, the corresponding subshift is called sofic. In particular, if X is finite then S is called a
subshift of finite type.

Examples

The first trivial example of shift space (of finite type) is the full shift j^ .

Let A = {a, b\ ■ The set of all infinite words over A containing at most one b is a sofic subshift, not of finite type.

Shift space 116

Further reading

• Lind, Douglas; Marcus, Brian (1995). An Introduction to Symbolic Dynamics and Coding. Cambridge UK:
Cambridge University Press. ISBN 0521559006.

• Lothaire, M. (2002). "Finite and Infinite Words" . Algebraic Combinatorics on Words . Cambridge UK:
Cambridge University Press. ISBN 0521812208. Retrieved 2008-01-29.

• Morse, Marston; Hedlund, Gustav A. (1938). "Symbolic Dynamics" (JSTOR). American Journal of Mathematics
60 (4): 815-866. doi: 10.2307/237 1264. JSTOR 2371264.

References

[1] Thomsen, K. (2004). "On the structure of a sofic shift space" (http://www.imf.au.dk/publications/pp/2003/imf-pp-2003-6.pdf) (PDF

Reprint). Transactions of the American Mathematical Society 356: 3557-3619. doi: 10. 1090/S0002-9947-04-03437-3. . Retrieved 2008-01-29.
[2] http://www-igm.univ-mlv.fr/%7Eberstel/Lothaire/ChapitresACW/Cl.ps
[3] http://www-igm.univ-mlv.fr/~berstel/Lothaire/AlgCWContents.html

Markov partition

A Markov partition is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be
applied to the study of hyperbolic systems. By using a Markov partition, the system can be made to resemble a
discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov
shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov
property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the
computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm
determinants and the like.

Motivation

Let (M,q>) be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic
representation: a faithful encoding of the points of M by sequences of symbols such that the map q> becomes the
shift map.

Suppose that M has been divided into a number of pieces E ,E ',..., E , which are thought to be as small and
localized, with virtually no overlaps. The behavior of a point x under the iterates of q> can be tracked by recording,
for each n, the part E. which contains q> (x). This results in an infinite sequence on the alphabet {l,2,...r} which
encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different
points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which
are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M
becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of
finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the
dynamical system (M,q>).

Markov partition 117

Formal definition

A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles
{E U E 2 , ■■ -E r ] such that

• For any pair of points x, y G Ei , that W B (x) D W u (y) G Ei

• h&Ei l~l IntEj = for i ^ j

• If x G IntEiand 0(x) G Lit .E,-, then

[W^) n Ei] D W u {<j)x) n £,-

Here, W^f^land W^fxjare the unstable and stable manifolds of x, respectively, and Intii^ simply denotes the

interior of Ei ■

These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that
is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and
not the past history of the system. It is this property of the covering that merits the 'Markov' appellation. The
resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968)
and Rufus Bowen (1975), thus putting symbolic dynamics on a firm footing.

Examples

Markov partitions have been constructed in several situations.

• Anosov diffeomorphisms of the torus.

• Dynamical billiards, in which case the covering is countable.

Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe.

References

[1] Pierre Gaspard, Chaos, scattering and statistical mechanics, (1998) Cambridge University Press.

• Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press,
1995 ISBN 0-521-55124-2

Sharkovskii's theorem 118

Sharkovskii's theorem

In mathematics, Sharkovskii's theorem, named after Oleksandr Mikolaiovich Sharkovsky, is a result about discrete
dynamical systems. One of the implications of the theorem is that if a continuous discrete dynamical system on the
real line has a periodic point of period 3, then it must have periodic points of every other period.

The theorem

Suppose

/:R->R

is a continuous function. We say that the number x is a periodic point of period m iff m (x) = x (where f m denotes the
composition of m copies off) and having least period m if furthermore/ (x) * x for all < k < m. We are interested
in the possible periods of periodic points off. Consider the following ordering of the positive integers:

3 5 7 9 11

.. (2n + l)-

■2°

3-2 5-2 7-2 9-2 11-2 .

.. (2n + l)-

■2 1

3-2 2 5-2 2 7-2 2 9-2 2 11 • 2 2 .

.. (2n + l).

•2 2

3-2 3 5-2 3 7-2 3 9-2 3 11 ■ 2 3 .

.. (2n + l)-

•2 3

T ... 2 5 2 4 2 3 2 1

We start, that is, with the odd numbers in increasing order, then 2 times the odds, 4 times the odds, 8 times the odds,

etc., and at the end we put the powers of two in decreasing order. Every positive integer appears exactly once
somewhere on this list. Sharkovskii's theorem states that if/ has a periodic point of least period m and m precedes n
in the above ordering, then/has also a periodic point of least period n.

As a consequence, we see that if/ has only finitely many periodic points, then they must all have periods which are
powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other
periods.

Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those
periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which
apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and
therefore not visible on the computer generated picture.

Interestingly, the above "Sharkovskii ordering" of the positive integers also occurs in a slightly different context in
connection with the logistic map: the stable cycles appear in this order in the bifurcation diagram, starting with 1 and
ending with 3, as the parameter is increased. (Here we ignore a stable cycle if a stable cycle of the same order has
occurred earlier.)

The assumption of continuity is important, as the discontinuous function f : x \ — ► (1 — x) _1 > f° r which every
value has period 3, would otherwise be a counterexample.

Sharkovskii's theorem 119

Generalizations

Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to
find a circle map with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some
generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit.

References

• Weisstein, Eric W., "Sharkovskys Theorem from MathWorld.

• Sarkovskii's theorem on PlanetMath

External link

• Keith Burns and Boris Hasselblatt, The Sharkovsky theorem: a natural direct proof

References

[1] http://mathworld.wolfram.com/SharkovskysTheorem.html

[2] http://planetmath.org/?op=getobj&from=objects&id=3751

[3] http://math.arizona.edu/~dwang/BurnsHasselblattRevised-l.pdf

Ergodic system

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related
problems. Its initial development was motivated by problems of statistical physics.

A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time.
The first result in this direction is the Poincare recurrence theorem, which claims that almost all points in any subset
of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems
which assert that, under certain conditions, the time average of a function along the trajectories exists almost
everywhere and is related to the space average. Two of the most important examples are ergodic theorems of
Birkhoff and von Neumann. For the special class of ergodic systems, the time average is the same for almost all
initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger
properties, such as mixing and equidistribution, have also been extensively studied.

The problem of metric classification of systems is another important part of the abstract ergodic theory. An
outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of
entropy for dynamical systems.

The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying
idea is that for certain systems the time average of their properties is equal to the average over the entire space.
Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for
systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on
Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature.
Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections
with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the
theory of diophantine approximations, L-functions).

Ergodic system 120

Ergodic transformations

Ergodic theory is often concerned with ergodic transformations.

Let T: X — > X be a measure-preserving transformation on a measure space (X, Z, fi), with u(X) = 1. A
measure-preserving transformation T as above is ergodic if for every
E G Swith T- X {E) = E either ^(E) = Oor p{E) = 1.

Examples

• An irrational rotation of the circle R/Z, T: x — > x+6*, where is irrational, is ergodic. This transformation has even
stronger properties of unique ergodicity, minimality, and equidistribution. By contrast, if 6 = plq is rational (in
lowest terms) then T is periodic, with period q, and thus cannot be ergodic: for any interval / of length a, < a <
\lq, its orbit under T is a T-invariant mod set that is a union of q intervals of length a, hence it has measure qa
strictly between and 1 .

• Let G be a compact abelian group, (i the normalized Haar measure, and T a group automorphism of G. Let G be
the Pontryagin dual group, consisting of the continuous characters of G, and T be the corresponding adjoint
automorphism of G . The automorphism T is ergodic if and only if the equality (T ) (x)-X i s possible only when n
= orx is the trivial character of G. In particular, if G is the n-dimensional torus and the automorphism T is
represented by an integral matrix A then T is ergodic if and only if no eigenvalue of A is a root of unity.

• A Bernoulli shift is ergodic. More generally, ergodicity of the shift transformation associated with a sequence of
i.i.d. random variables and some more general stationary processes follows from Kolmogorov's zero-one law.

• Ergodicity of a continuous dynamical system means that its trajectories "spread around" the phase space. A
system with a compact phase space which has a non-constant first integral cannot be ergodic. This applies, in
particular, to Hamiltonian systems with a first integral / functionally independent from the Hamilton function H
and a compact level setX= {(p,q): H(p,q)=E] of constant energy. Liouville's theorem implies the existence of a
finite invariant measure on X, but the dynamics of the system is constrained to the level sets of / on X, hence the
system possesses invariant sets of positive but less than full measure. A property of continuous dynamical
systems that is the opposite of ergodicity is complete integrability.

Ergodic theorems

Let T : X — > X be a measure-preserving transformation on a measure space (X, S, u) ■ One may then consider
the "time average" of a fJ> -integrable function/, i.e. f £ L^(ix)- The "time average" is defined as the average (if
it exists) over iterations of T starting from some initial point x.

/W = Ja -E/(i**)-

" fc=0
If u(X) is finite and nonzero, we can consider the "space average" or "phase average" off, defined as

1 f
f = I f dfj,. (For a probability space, fi(X) = 1.)

In general the time average and space average may be different. But if the transformation is ergodic, and the measure
is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic
theorem, in an abstract form due to George David Birkhoff. (Actually, Birkhoff s paper considers not the abstract
general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The
equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of
probabilities on the unit interval.

More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of
/exists for almost every x and that the (almost everywhere defined) limit function f is integrable:

Ergodic system 121

Furthermore, f is T-invariant, that is to say

foT = f

holds almost everywhere, and if u,(X) is finite, then the normalization is the same:

J fdfi = J fdfi.

In particular, if T is ergodic, then f must be a constant (almost everywhere), and so one has that

/=/

almost everywhere. Joining the first to the last claim and assuming that u,(X) is finite and nonzero, one has that

1 n ~ ^ 1 f

lim -Y f (T k x) = — — / / du

for almost all x, i.e., for all x except for a set of measure zero.

For an ergodic transformation, the time average equals the space average almost surely.

As an example, assume that the measure space (X, S,u) models the particles of a gas as above, and let f(x)

denotes the velocity of the particle at position x. Then the pointwise ergodic theorems says that the average velocity

of all particles at some given time is equal to the average velocity of one particle over time.

Probabilistic formulation: Birkhoff— Khinchin theorem

Birkhoff— Khinchin theorem. Let /be measurable, i£(|/|) < +oo , and J'be a measure-preserving map. Then

lim -^/(rV)=£;(/|c)a.s,

k=0
where E(f\C)is> the conditional expectation given the a -algebra Q of invariant sets of J" 1 -

Corollary (Pointwise ergodic theorem) In particular, if J 1 is also ergodic, then Q is the trivial a -algebra, and thus

- n— 1

lim -V/(T fe a :)=£(/)a. S .

n — too n. ' *

Ti— >CM n

fe=0

Mean ergodic theorem

Another form of the ergodic theorem, von Neumann's mean ergodic theorem, holds in Hilbert spaces.

Let JJ be a unitary operator on a Hilbert space ff ; more generally, an isometric linear operator (that is, a not

necessarily surjective linear operator satisfying II £7x|| = llxll for all x £ H > or equivalently, satisfying

U* U = I, but not necessarily [/"[/* =/)■ Let pbe the orthogonal projection onto

ty G i?|C7^ = ^} = Ker(7 - 27).
Then, for any x G i7 . we have:

i JV-1
lim — V U n x = Px,

Th=\}

where the limit is with respect to the norm on H. In other words, the sequence of averages

1 N-l
converges to P in the strong operator topology.

This theorem specializes to the case in which the Hilbert space H consists of L functions on a measure space and U
is an operator of the form

Ergodic system 122

Uf(x) = f(Tx)
where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a
discrete dynamical system. The ergodic theorem then asserts that the average behavior of a function / over
sufficiently large time-scales is approximated by the orthogonal component off which is time-invariant.

In another form of the mean ergodic theorem, let U be a strongly continuous one-parameter group of unitary
operators on H. Then the operator

U t dt

-f 1

T Jo

converges in the strong operator topology as T — > °°. In fact, this result also extends to the case of strongly
continuous one-parameter semigroup of contractive operators on a reflexive space.

Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex
numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we
pick a single complex number of unit length (which we think of as JJ ), it is intuitive that its powers will fill up the
circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of JJ will converge to
0. Also, is the only fixed point of JJ , and so the projection onto the space of fixed points must be the zero
operator (which agrees with the limit just described).

Convergence of the ergodic means in the //norms

Let (X, S, u)be as above a probability space with a measure preserving transformation T, and let 1 < p < oo .
The conditional expectation with respect to the sub-o-algebra S r of the T-invariant sets is a linear projector £J r of
norm 1 of the Banach space LPiX, S, u)onto its closed subspace L P (X, Ey, /i).The latter may also be
characterized as the space of all T-invariant £JP -functions on X. The ergodic means, as linear operators on
LPiX, S, u)also have unit operator norm; and, as a simple consequence of the Birkhoff— Khinchin theorem,
converge to the projector E T in the strong operator topology of £JP if 1 < p < oo,and in the weak operator
topology if p = OO . More is true if 1 < p < OO :then the Wiener— Yoshida— Kakutani ergodic dominated
convergence theorem states that the ergodic means of f £ Z/are dominated in JJ 1 ; however, if f (= ]_}-, the
ergodic means may fail to be equidominated in jJ-. Finally, if/ is assumed to be in the Zygmund class, that is
I f\ log + I f I is integrable, then the ergodic means are even dominated in j^-.

Sojourn time

Let (X, S, u) be a measure space such that u,(X) is finite and nonzero. The time spent in a measurable set A is
called the sojourn time. An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative
measure of A is equal to the mean sojourn time:

KA) _ i r ,.._,„ i- 1

/ Xa dp = lim - V xa (T k x)

for all x except for a set of measure zero, where XA is the indicator function of A.

Let the occurrence times of a measurable set A be defined as the set k , k , k , ..., of times k such that i(x) is in A,

sorted in increasing order. The differences between consecutive occurrence times R = k. - k , are called the

i i i—i

recurrence times of A. Another consequence of the ergodic theorem is that the average recurrence time of A is
;ly proportional to the measure

inversely proportional to the measure of A, assuming that the initial point x is in A, so that k =

(almost surely)

n fi(A)

(See almost surely.) That is, the smaller A is, the longer it takes to return to it.

Ergodic system 123

Ergodic flows on manifolds

The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact
manifolds of constant negative curvature of any dimension was proved by Eberhard Hopf in 1939, although special
cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924). The relation
between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2,R) was described in 1952 by S.
V. Fomin and I. M. Gelfand. The article on Anosov flows provides an example of ergodic flows on SL(2,R) and on
Riemann surfaces of negative curvature. Much of the development described there generalizes to hyperbolic
manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in the semisimple
Lie group SO(n,l). Ergodicity of the geodesic flow on Riemannian symmetric spaces was demonstrated by F. I.
Mautner in 1957. In 1967 D. V. Anosov and Ya. G. Sinai proved ergodicity of the geodesic flow on compact
manifolds of variable negative sectional curvature. A simple criterion for the ergodicity of a homogeneous flow on a
homogeneous space of a semisimple Lie group was given by Calvin C. Moore in 1966. Many of the theorems and
results from this area of study are typical of rigidity theory.

In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic.
Unique ergodicity of the flow was established by Hillel Furstenberg in 1972. Ratner's theorems provide a major
generalization of ergodicity for unipotent flows on the homogeneous spaces of the form AG, where G is a Lie group
and r is a lattice in G.

In the last 20 years, there have been many works trying to find a measure-classification theorem similar to Ratner's
theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis. An important partial
result (solving those conjectures with an extra assumption of positive entropy) was proved by Elon Lindenstrauss,
and he was awarded the Fields medal in 2010 for this result.

References

1] I: Functional Analysis : Volume 1 by Michael Reed, Barry Simon.Academic Press; REV edition (1980)
2] (Walters 1982)

Historical references

Birkhoff, George David (1931), "Proof of the ergodic theorem" (http://www.pnas.org/cgi/reprint/17/12/656),

Proc Natl Acad Sci USA 17 (12): 656-660, doi: 10. 1073/pnas. 17. 12.656, PMC 1076138, PMID 16577406.

Birkhoff, George David (1942), "What is the ergodic theorem?", American Mathematical Monthly (The American

Mathematical Monthly, Vol. 49, No. 4) 49 (4): 222-226, doi: 10.2307/2303229, JSTOR 2303229.

von Neumann, John (1932), "Proof of the Quasi-ergodic Hypothesis", Proc Natl Acad Sci USA 18 (1): 70—82,

doi:10.1073/pnas.l8.1.70, PMC 1076162, PMID 16577432.

von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA 18 (3):

263-266, doi:10.1073/pnas.l8.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.

Hopf, Eberhard (1939), "Statistik der geodatischen Linien in Mannigfaltigkeiten negativer Krummung", Leipzig

Ber. Verhandl. Sachs. Akad. Wiss. 91: 261—304.

Fomin, Sergei V.; Gelfand, I. M. (1952), "Geodesic flows on manifolds of constant negative curvature", Uspehi

Mat.Naukl (1): 118-137.

Mautner, F. I. (1957), "Geodesic flows on symmetric Riemann spaces", Ann. Of Math. (The Annals of

Mathematics, Vol. 65, No. 3) 65 (3): 416-431, doi: 10.2307/1970054, JSTOR 1970054.

Moore, C. C. (1966), "Ergodicity of flows on homogeneous spaces", Amer. J. Math. (American Journal of

Mathematics, Vol. 88, No. 1) 88 (1): 154-178, doi: 10.2307/2373052, JSTOR 2373052.

Ergodic system 124

Modern references

D.V. Anosov (2001), "Ergodic theory" (http://eom.springer.de/e/e036150.htm), in Hazewinkel, Michiel,

Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

This article incorporates material from ergodic theorem on PlanetMath, which is licensed under the Creative

Commons Attribution/Share-Alike License.

Vladimir Igorevich Arnol'd and Andre Avez, Ergodic Problems of Classical Mechanics. New York: W.A.

Benjamin. 1968.

Leo Breiman, Probability. Original edition published by Addison— Wesley, 1968; reprinted by Society for

Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. (See Chapter 6.)

Peter Walters, An introduction to ergodic theory, Springer, New York, 1982, ISBN 0-387-95152-0.

Tim Bedford, Michael Keane and Caroline Series, eds. (1991), Ergodic theory, symbolic dynamics and hyperbolic

spaces, Oxford University Press, ISBN 0-19-853390-X (A survey of topics in ergodic theory; with exercises.)

Karl Petersen. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge

University Press. 1990.

Joseph M. Rosenblatt and Mate Weirdl, Pointwise ergodic theorems via harmonic analysis, (1993) appearing in

Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference,

(1995) Karl E. Petersen and Ibrahim A. Salama, eds., Cambridge University Press, Cambridge, ISBN

0-521-45999-0. (An extensive survey of the ergodic properties of generalizations of the equidistribution theorem

of shift maps on the unit interval. Focuses on methods developed by Bourgain.)

• A.N. Shiryaev, Probability, 2nd ed., Springer 1996, Sec. V.3. ISBN 0-387-94549-0.

External links

• Ergodic Theory (29 October 2007) (http://www.cscs.umich.edu/~crshalizi/notebooks/ergodic-theory.html)
Notes by Cosma Rohilla Shalizi

Ergodic theory 125

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related
problems. Its initial development was motivated by problems of statistical physics.

Ergodic transformations

Ergodic theory is often concerned with ergodic transformations.

Let T: X — » X be a measure-preserving transformation on a measure space (X, 1, fi), with u(X) = 1. A
measure-preserving transformation T as above is ergodic if for every
E G Swith T- X {E) = £ either ^E) = Oor p{E) = 1.

Examples

• An irrational rotation of the circle R/Z, T: x — > x+9, where 6 is irrational, is ergodic. This transformation has even
stronger properties of unique ergodicity, minimality, and equidistribution. By contrast, if 6 = plq is rational (in
lowest terms) then T is periodic, with period q, and thus cannot be ergodic: for any interval / of length a, < a <
l/q, its orbit under T is a T-invariant mod set that is a union of q intervals of length a, hence it has measure qa
strictly between and 1 .

• Let G be a compact abelian group, fj. the normalized Haar measure, and T a group automorphism of G. Let G be
the Pontryagin dual group, consisting of the continuous characters of G, and T be the corresponding adjoint
automorphism of G . The automorphism T is ergodic if and only if the equality (T ) (x)-X i s possible only when n
= or^ is the trivial character of G. In particular, if G is the « -dimensional torus and the automorphism T is
represented by an integral matrix A then T is ergodic if and only if no eigenvalue of A is a root of unity.

Ergodic theory 126

Ergodic theorems

Let T \ X > X^ e a measure-preserving transformation on a measure space (X, S, ji) ■ One may then consider

the "time average" of a \l -integrable function/, i.e. f (= L^(ll)- The "time average" is defined as the average (if
it exists) over iterations of T starting from some initial point x.

/» = il- -£/(***)■

" fc=0
If u(X) is finite and nonzero, we can consider the "space average" or "phase average" off, defined as

1 f
f = I f dfi. (For a probability space, fJ,(X) = 1.)

M^O J

In general the time average and space average may be different. But if the transformation is ergodic, and the measure

is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic
theorem, in an abstract form due to George David Birkhoff. (Actually, Birkhoff s paper considers not the abstract
general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The
equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of
probabilities on the unit interval.

Furthermore, f is T-invariant, that is to say

foT = f

holds almost everywhere, and if u(X) is finite, then the normalization is the same:

J fdfi = J fdfi.

In particular, if T is ergodic, then f must be a constant (almost everywhere), and so one has that

/=/

almost everywhere. Joining the first to the last claim and assuming that u(X) is finite and nonzero, one has that

1 n ~ ^ 1 f

for almost all x, i.e., for all x except for a set of measure zero.

For an ergodic transformation, the time average equals the space average almost surely.

As an example, assume that the measure space (X, S,u) models the particles of a gas as above, and let f(x)

denotes the velocity of the particle at position x. Then the pointwise ergodic theorems says that the average velocity

of all particles at some given time is equal to the average velocity of one particle over time.

Ergodic theory 127

Probabilistic formulation: Birkhoff— Khinchin theorem

Birkhoff— Khinchin theorem. Let fbe measurable, E(\f I) < +OO , and J/be a measure-preserving map. Then
lim - ^ / (^x) = £(/|C) a.s.

k=0

where _EY/|C)is the conditional expectation given the u -algebra Q of invariant sets of J 1 -

Corollary (Pointwise ergodic theorem) In particular, if J 1 is also ergodic, then Q is the trivial a -algebra, and thus

- n— 1

lim -V/(T fe x)=£(./)a. S .

fe=0

Mean ergodic theorem

Another form of the ergodic theorem, von Neumann's mean ergodic theorem, holds in Hilbert spaces.

Let JJ be a unitary operator on a Hilbert space JJ ; more generally, an isometric linear operator (that is, a not

necessarily surjective linear operator satisfying II £7x|| = llxll for all x£iJ, or equivalently, satisfying

U*U = 7, but not necessarily JJU* =7)- Let pbe the orthogonal projection onto

{t/> G 77|C7^ = iP} = Ker(7 - 27).
Then, for any x £ i7 > we have:

i JV-1
lim — Y U n x = Px,

j^~> AT £—< '

Tfc=U

1 E^

^=0

where the limit is with respect to the norm on H. In other words, the sequence of averages

JV-1

converges to P in the strong operator topology.

This theorem specializes to the case in which the Hilbert space H consists of L functions on a measure space and U
is an operator of the form

In another form of the mean ergodic theorem, let U be a strongly continuous one-parameter group of unitary
operators on H. Then the operator

, r U t dt

rji I *>

converges in the strong operator topology as T — > °°. In fact, this result also extends to the case of strongly
continuous one-parameter semigroup of contractive operators on a reflexive space.

T Jo

Ergodic theory 128

Convergence of the ergodic means in the w norms

Let (X, S, u)be as above a probability space with a measure preserving transformation T, and let 1 < p < oo .
The conditional expectation with respect to the sub-o-algebra S r of the T-invariant sets is a linear projector £J r of
norm 1 of the Banach space LPiX, S, u)onto its closed subspace L P (X, Ey, /i).The latter may also be
characterized as the space of all T-invariant £JP -functions on X. The ergodic means, as linear operators on
LP(X, S, u)also have unit operator norm; and, as a simple consequence of the Birkhoff— Khinchin theorem,
converge to the projector E T in the strong operator topology of ^Pif 1 < p < 0O,and in the weak operator
topology if p = OO . More is true if 1 < p < oo :then the Wiener— Yoshida— Kakutani ergodic dominated
convergence theorem states that the ergodic means of f £ Z/are dominated in ]_?; however, if f (= X/ 1 . the
ergodic means may fail to be equidominated in jj-. Finally, if/ is assumed to be in the Zygmund class, that is
I f\ log + I f\ is integrable, then the ergodic means are even dominated in j^-.

Sojourn time

Let (X, S, ji) be a measure space such that n(X ) is finite and nonzero. The time spent in a measurable set A is
called the sojourn time. An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative
measure of A is equal to the mean sojourn time:

nvs = -Tin l XAd » = ^ i £^ ( Tkx )

H(X) fi{X) J <™ n ^ V )

for all x except for a set of measure zero, where XA is the indicator function of A.

Let the occurrence times of a measurable set A be defined as the set k , k , k , ..., of times k such that i{x) is in A,

sorted in increasing order. The differences between consecutive occurrence times R = k. - k , are called the

i i i-l

recurrence times of A. Another consequence of the ergodic theorem is that the average recurrence time of A is
;ly proportional to the measure

inversely proportional to the measure of A, assuming that the initial point x is in A, so that k =

, (almost surely)

n V{A)

(See almost surely.) That is, the smaller A is, the longer it takes to return to it.

Ergodic flows on manifolds

Ergodic theory 129

References

[1] I: Functional Analysis : Volume 1 by Michael Reed, Barry Simon, Academic Press; REV edition (1980)
[2] (Walters 1982)

Historical references

Birkhoff, George David (1931), "Proof of the ergodic theorem" (http://www.pnas.org/cgi/reprint/17/12/656),

Proc Natl Acad Sci USA 17 (12): 656-660, doi: 10. 1073/pnas. 17. 12.656, PMC 1076138, PMID 16577406.

Birkhoff, George David (1942), "What is the ergodic theorem?", American Mathematical Monthly (The American

Mathematical Monthly, Vol. 49, No. 4) 49 (4): 222-226, doi: 10.2307/2303229, JSTOR 2303229.

von Neumann, John (1932), "Proof of the Quasi-ergodic Hypothesis", Proc Natl Acad Sci USA 18 (1): 70-82,

doi:10.1073/pnas.l8.1.70, PMC 1076162, PMID 16577432.

von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA 18 (3):

263-266, doi:10.1073/pnas.l8.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.

Hopf, Eberhard (1939), "Statistik der geodatischen Linien in Mannigfaltigkeiten negativer Kriimmung", Leipzig

Ber. Verhandl. Sachs. Akad. Wiss. 91: 261—304.

Fomin, Sergei V.; Gelfand, I. M. (1952), "Geodesic flows on manifolds of constant negative curvature", Uspehi

Mat. Nauk 7(1): 118-137.

Mautner, F. I. (1957), "Geodesic flows on symmetric Riemann spaces", Ann. Of Math. (The Annals of

Mathematics, Vol. 65, No. 3) 65 (3): 416-431, doi: 10.2307/1970054, JSTOR 1970054.

Moore, C. C. (1966), "Ergodicity of flows on homogeneous spaces", Amer. J. Math. (American Journal of

Mathematics, Vol. 88, No. 1) 88 (1): 154-178, doi: 10.2307/2373052, JSTOR 2373052.

Modern references

D.V. Anosov (2001), "Ergodic theory" (http://eom.springer.de/e/e036150.htm), in Hazewinkel, Michiel,

Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

This article incorporates material from ergodic theorem on PlanetMath, which is licensed under the Creative

Commons Attribution/Share-Alike License.

Vladimir Igorevich Arnol'd and Andre Avez, Ergodic Problems of Classical Mechanics. New York: W.A.

Benjamin. 1968.

Leo Breiman, Probability. Original edition published by Addison— Wesley, 1968; reprinted by Society for

Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. (See Chapter 6.)

Peter Walters, An introduction to ergodic theory, Springer, New York, 1982, ISBN 0-387-95152-0.

Tim Bedford, Michael Keane and Caroline Series, eds. (1991), Ergodic theory, symbolic dynamics and hyperbolic

spaces, Oxford University Press, ISBN 0-19-853390-X (A survey of topics in ergodic theory; with exercises.)

Karl Petersen. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge

University Press. 1990.

Joseph M. Rosenblatt and Mate Weirdl, Pointwise ergodic theorems via harmonic analysis, (1993) appearing in

Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference,

(1995) Karl E. Petersen and Ibrahim A. Salama, eds., Cambridge University Press, Cambridge, ISBN

0-521-45999-0. (An extensive survey of the ergodic properties of generalizations of the equidistribution theorem

of shift maps on the unit interval. Focuses on methods developed by Bourgain.)

Ergodic theory 130

• A.N. Shiryaev, Probability, 2nd ed., Springer 1996, Sec. V.3. ISBN 0-387-94549-0.

External links

• Ergodic Theory (29 October 2007) (http://www.cscs.umich.edu/~crshalizi/notebooks/ergodic-theory.html)
Notes by Cosma Rohilla Shalizi

Measure-preserving dynamical system

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of
dynamical systems, and ergodic theory in particular.

Definition

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation
on it. In more detail, it is a system

(X, B, p, T)

with the following structure:

• J{" is a set,

• B is a o-algebra over X >

• 11 : B — > [0, 1] is a probability measure, so that IJ,(X) = 1, and

• T : X — > X i s a measurable transformation which preserves the measure fl< , i. e. each A £ B satisfies

p(T- 1 A) = p(A).

This definition can be generalized to the case in which J 1 is not a single transformation that is iterated to give the
dynamics of the system, but instead is a monoid (or even a group) of transformations T s : X — > X parametrized
by s £ Z ( or R > or N U {0} , or [0, +oo)), where each transformation T s satisfies the same requirements as
J 1 above. In particular, the transformations obey the rules

• To = idjf : X — > X , the identity function on X '■>

• T s o Tj = Ti _|_ s , whenever all the terms are well-defined;

• T 1- 1 — T 1 whenever all the terms are well-defined.

The earlier, simpler case fits into this framework by defining T s := T s for s £ N •

The existence of invariant measures for certain maps and Markov processes is established by the

Krylov— Bogolyubov theorem.

Measure-preserving dynamical system

131

Examples

Examples include:

• u, could be the normalized angle measure d6/2jt on the unit circle,
and J 1 a rotation. See equidistribution theorem;

• the Bernoulli scheme;

• the interval exchange transformation;

• with the definition of an appropriate measure, a subshift of finite
type;

• the base flow of a random dynamical system.

Homomorphisms

The concept of a homomorphism and an isomorphism may be defined.

T-HA)

Example of a (Lebesgue) measure preserving
map: T l [0, 1) -► [0, 1),

x i— > 2x mod 1.

Consider two dynamical systems (X, jA., U, X")and (Y, B, V, S)- Then a mapping

<f>:X ^Y
is a homomorphism of dynamical systems if it satisfies the following three properties:

1 . The map cp is measurable,

2. For each B £ B , one has fJ ,((jr 1 B) = v{B),

3. For u-almost all x € X> one nas <p(Tx) — S((j)x)-

The system (Y, B, V, 5) is then called a factor of [X, A, /i, T).

The map (p is an isomorphism of dynamical systems if, in addition, there exists another mapping

iP:Y -^X

that is also a homomorphism, which satisfies

1. For u-almost all x £ X . one nas x — lj)((f)x)

2. For v-almost all y £ Y , one has y = (j){tj)y\.

Generic points

A point j^Xis called a generic point if the orbit of the point is distributed uniformly according to the measure.

Symbolic names and generators

Consider a dynamical system (X, B, T, Li), and let Q = { Q , ..., Q } be a partition of X into k measurable
pair-wise disjoint pieces. Given a point xGX, clearly x belongs to only one of the Q.. Similarly, the iterated point
T x can belong to only one of the parts as well. The symbolic name of x, with regards to the partition Q, is the
sequence of integers {a } such that

T n x£Q an . "

The set of symbolic names with respect to a partition is called the symbolic dynamics of the dynamical system. A
partition Q is called a generator or generating partition if [i-almost every point x has a unique symbolic name.

Measure-preserving dynamical system 132

Operations on partitions

Given a partition Q - { Q , ..., Q } and a dynamical system (X, 3, T, Li) , we define T-pullback of Q as

T- 1 Q = {T- 1 Q 1 ,...,T~ 1 Q k }.

Further, given two partitions Q = { Q , ..., Q } and R = { R , ..., R }, we define their refinement QvR as

QVR={Q i nR j \i = l,...,k, j = l,...,m, /x(Qi n J2,-) > 0}.

With these two constructs we may define refinement of an iterated pullback

^=oT~ n Q = {Q i0 n T-'Qi, n ■ ■ - n T"^

|*, = 1,...,A;, £ = 0,...,iV,

/.(g i0 n T" 1 ^, n ■ ■ ■ n T~ N Q iN ) > 0}

which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.

Measure-theoretic entropy

The entropy of a partition Q is defined as
k.

i7(Q) = -X>(Q m )log/i(Q m ).

771=1

The measure-theoretic entropy of a dynamical system Of ; 3 T, Li) with respect to a partition 2 = { Q , ..., Q } is
then defined as

MT,Q)= J lim^(V/-0)-

\Tl=U /

Finally, the Kolmogorov— Sinai or measure-theoretic entropy of a dynamical system (X, 3, T, uV s defined as

/i^(!T) = sup^(T, Q).
Q

where the supremum is taken over all finite measurable partitions. A theorem of Yakov G. Sinai in 1959 shows that

the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli
process is log 2, since every real number has a unique binary expansion. That is, one may partition the unit interval
into the intervals [0, 1/2) and [1/2, 1]. Every real number x is either less than 1/2 or not; and likewise so is the
fractional part of 2 x.

If the space X is compact and endowed with a topology, or is a metric space, then the topological entropy may also
be defined.

References

• Michael S. Keane, Ergodic theory and subshifts of finite type, (1991), appearing as Chapter 2 in Ergodic Theory,
Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford
University Press, Oxford (1991). ISBN 0-19-853390-X (Provides expository introduction, with exercises, and
extensive references.)

• Lai-Sang Young, "Entropy in Dynamical Systems" (pdf ; ps ), appearing as Chapter 16 in Entropy, Andreas
Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003). ISBN
0-691-11338-6

Measure-preserving dynamical system

133

Examples

• T. Schiirmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A28, page 5033ff,

1995. PDF-Dokument

[3]

References

[1] http://www.math.nyu.edu/~lsy/papers/entropy.pdf
[2] http://www.math.nyu.edu/~lsy/papers/entropy.ps
[3] http://arxiv.org/abs/nlin/0208048

Periodic orbit

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function
of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase
space, that is different orbits do not intersect in the phase space. Understanding the properties of orbits by using
topological method is one of the objectives of the modern theory of dynamical systems.

For discrete-time dynamical systems the orbits are sequences, for real dynamical systems the orbits are curves and
for holomorphic dynamical systems the orbits are Riemann surfaces.

Definition

Given a dynamical system (T, M, O) with T
a group, M a set and <J> the evolution
function

Real Space

\ \

Phase Space

Orbit

1 Velocity

Diagram showing the periodic orbit of a mass-spring system in simple harmonic

motion. (Here the velocity and position axes have been reversed from the standard

convention in order to align the two diagrams)

$ : U -> M where U CTxM
we define

/(!):={££!: (t,x) ££/},
then the set

7, :={$(*, x):i€/(x)}

Periodic orbit 1 34

is called orbit through x. An orbit which consists of a single point is called constant orbit. A non-constant orbit is
called closed or periodic if there exists a t in T so that

<&(£, x) — x
for every point x on the orbit.

Real dynamical system

Given a real dynamical system (R, M, O), I{x) is an open interval in the real numbers, that is I(x) = (t~ £"*")• F° r

any x in M

7+:= {*(*,*) :f€(0,t+)}
is called positive semi-orbit through x and

7-:= {*(*,*):*€(£, 0)}
is called negative semi-orbit through x.

Discrete time dynamical system

For discrete time dynamical system :
forward orbit of x is a set :

7+ = {Q(t,z) : * > 0}
backward orbit of x is a set :

-Y- = W-t,x):t>0}

and orbit of x is a set :

def _ _|_

where :

• <|>is an evolution function <§> \ X — > X which is here an iterated function,

• set X i s dynamical space,

• i is number of iteration, which is natural number and t €z T

• xis initial state of system and x £ X
Usually different notation is used :

• <E>(i, a;)is noted as ^(x)

• Xf = $*(x)with Xqis a x from above notation.

Periodic orbit

135

Notes

It is often the case that the evolution function can be understood to compose the elements of a group, in which case
the group-theoretic orbits of the group action are the same thing as the dynamical orbits.

Examples

• The orbit of an equilibrium point is a constant orbit

Stability of orbits

A basic classification of orbits is

• constant orbits or fixed points

• periodic orbits

• non-constant and non-periodic orbits

An orbit can fail to be closed in two ways. It could be an

asymptotically periodic orbit if it converges to a periodic orbit. Such

orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit. An orbit

can also be chaotic. These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic

orbit. They exhibit sensitive dependence on initial conditions, meaning that small differences in the initial value will

cause large differences in future points of the orbit.

There are other properties of orbits that allow for different classifications. An orbit can be hyperbolic if nearby points
approach or diverge from the orbit exponentially fast.

Critical orbit of discrete dynamical system based

on complex quadratic polynomial. It tends to

weakly attracting fixed point with

multiplier=0.99993612384259

References

• Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge.
ISBN 0-521-57557-5.

Hilbert space

136

Hilbert space

Hilbert spaces can be used to study the harmonics
of vibrating strings.

The mathematical concept of a Hilbert space, named after David
Hilbert, generalizes the notion of Euclidean space. It extends the
methods of vector algebra and calculus from the two-dimensional
Euclidean plane and three-dimensional space to spaces with any finite
or infinite number of dimensions. A Hilbert space is an abstract vector
space possessing the structure of an inner product that allows length
and angle to be measured. Furthermore, Hilbert spaces are required to
be complete, a property that stipulates the existence of enough limits in
the space to allow the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics, physics,

and engineering, typically as infinite-dimensional function spaces. The

earliest Hilbert spaces were studied from this point of view in the first

decade of the 20th century by David Hilbert, Erhard Schmidt, and

Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics,

Fourier analysis (which includes applications to signal processing and heat transfer) and ergodic theory which forms

the mathematical underpinning of the study of thermodynamics. John von Neumann coined the term "Hilbert space"

for the abstract concept underlying many of these diverse applications. The success of Hilbert space methods ushered

in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces

include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized

functions, and Hardy spaces of holomorphic functions.

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the
Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto
a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and
other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to
a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of
axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite
sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good
cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions
in a sense that is made precise by the study of their spectrum.

Definition and illustration

Motivating example: Euclidean space

One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors,

denoted by R , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real
number xy. If x and y are represented in Cartesian coordinates, then the dot product is defined by

(xi, x 2 , x 3 ) ■ (y 1: y 2 , yz) = zij/i + x 2 y 2 + x 3 y 3 .
The dot product satisfies the properties:

1 . It is symmetric in x and y: xy = yx.

2. It is linear in its first argument: (ax + bx )-y = ax -y + bx -y for any scalars a, b, and vectors x , x , and y.

3. It is positive definite: for all vectors x, xx > with equality if and only if x = 0.

Hilbert space

137

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner
product. A vector space equipped with such an inner product is known as a (real) inner product space. Every
finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it
with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted llxll, and to the angle 6
between two vectors x and y by means of the formula

x ■ y = ||x|| ||y|| cos8.
Multivariable calculus in Euclidean space relies on the ability to
compute limits, and to have useful criteria for concluding that limits
exist. A mathematical series

Completeness means that if a particle moves

along the broken path (in blue) travelling a finite

total distance, then the particle has a well-defined

net displacement (in orange).

71=0

consisting of vectors in R is absolutely convergent provided that the sum of the lengths converges as an ordinary
series of real numbers:

5^||Xjfc|| < OO.

fc=0
Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in

the Euclidean space, in the sense that

L — 2_. x fe - > as JV — j- oo.

This property expresses the completeness of Euclidean space: that a series which converges absolutely also
converges in the ordinary sense.

Definition

A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the

T21
distance function induced by the inner product. To say that H is a complex inner product space means that H is a

complex vector space on which there is an inner product (x,y) associating a complex number to each pair of elements

x,y of H that satisfies the following properties:

• (y,;c) is the complex conjugate of (x,y):

[3]

(y,x) = (x,y).
(x,y) is linear in its first argument. LJJ For all complex numbers a and b,

(ax 1 + bx 2 ,y) = a{x u y) + b(x 2 ,y).
The inner product (•,•) is positive definite:

(x,x) >
where the case of equality holds precisely when x = 0.

Hilbert space

138

It follows from properties 1 and 2 that a complex inner product is antilinear in its second argument, meaning that

(x,ay 1 + by 2 ) = d(x,y 1 ) + b(x,y 2 ).
A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes
real values. Such an inner product will be bilinear: that is, linear in each argument.

The norm defined by the inner product (•,•) is the real-valued function

||x|| = \J(x,x),
and the distance between two points x,y in H is defined in terms of the norm by

d(x,y) = \\x-y\\ = y [x - y, x - y).
That this function is a distance function means (1) that it is symmetric in x and y, (2) that the distance between x and
itself is zero, and otherwise the distance between x and y must be positive, and (3) that the triangle inequality holds,
meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs:

d{x,z) < d(x,y) + d(y,z).

This last property is ultimately a consequence of the more fundamental Cauchy— Schwarz inequality, which asserts

\(x,y)\ < \\x\\ \\y\\

with equality if and only if x and y are linearly dependent.

Relative to a distance function defined in this way, any inner product space is a metric space, and sometimes is

known as a pre-Hilbert space. A pre-Hilbert space is a Hilbert space if in addition it is complete. Completeness is
expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H is complete if every
Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized
by the following equivalent condition: if a series of vectors XlfcLo u k conver g es absolutely in the sense that

X^IKII < °°!

fc=0
then the series converges in H, in the sense that the partial sums converge to an element of H.

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological
vector spaces, in which topological notions like the openness and closedness of subsets are well-defined. Of special
importance is the notion of a closed linear subspace of a Hilbert space which, with the inner product induced by
restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own
right.

Hilbert space

139

Second example: sequence spaces

The sequence space D consists of all infinite sequences z = (z ,z ,...) of complex numbers such that the series
|2

converges. The inner product on D is defined by

(z, w) = ^ Z n W^,,

71=1

with the latter series converging as a consequence of the Cauchy— Schwarz inequality.

Completeness of the space holds provided that whenever a series of elements from D converges absolutely (in norm),

then it converges to an element of D . The proof is basic in mathematical analysis, and permits mathematical series of
elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a

finite-dimensional Euclidean space)

[5]

History

Prior to the development of Hilbert spaces, other generalizations of
Euclidean spaces were known to mathematicians and physicists. In
particular, the idea of an abstract linear space had gained some traction
towards the end of the 19th century: this is a space whose elements
can be added together and multiplied by scalars (such as real or
complex numbers) without necessarily identifying these elements with
"geometric" vectors, such as position and momentum vectors in
physical systems. Other objects studied by mathematicians at the turn
of the 20th century, in particular spaces of sequences (including series)
and spaces of functions, can naturally be thought of as linear spaces.
Functions, for instance, can be added together or multiplied by
constant scalars, and these operations obey the algebraic laws satisfied
by addition and scalar multiplication of spatial vectors.

In the first decade of the 20th century, parallel developments led to the
introduction of Hilbert spaces. The first of these was the observation,
which arose during David Hilbert and Erhard Schmidt's study of

ro]

integral equations, that two square-integrable real-valued functions/
and g on an interval [a,b] have an inner product

(/, 9) = / f{^)g{x) dx

which has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal
family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to
prove an analog of the spectral decomposition for an operator of the form

f(x) i-> / K(x, y)f{y) dy

J a

where K is a continuous function symmetric in x and y. The resulting eigenfunction expansion expresses the function
K as a series of the form

K i x -,y) = ^2\*<p n {x)<pM

Hilbert space 140

where the functions w are orthogonal in the sense that (w ,w ) = for all n * m. The individual terms in this

n n m

series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions which
fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures
convergence, is completeness.

The second development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri
Lebesgue in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907,

Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space L of square Lebesgue-integrable
functions is a complete metric space. As a consequence of the interplay between geometry and completeness, the
19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily
carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as
the Riesz-Fischer theorem.

Further basic results were proved in the early 20th century. For example, the Riesz representation theorem was

ri3i

independently established by Maurice Frechet and Frigyes Riesz in 1907. John von Neumann coined the term

ri4i

abstract Hilbert space in his work on unbounded Hermitian operators. Although other mathematicians such as
Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a
physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von
Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued
work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in

ri7i

his book on quantum mechanics and the theory of groups.

The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best
mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are
vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the
system are unitary operators, and measurements are orthogonal projections. The relation between quantum

mechanical symmetries and unitary operators provided an impetus for the development of the unitary representation

ri7i
theory of groups, initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became

clear that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the

ri9i

framework of ergodic theory.

The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space,
according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began
investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by
von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark
Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand
made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the
operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that
underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic
in abstract harmonic analysis and representation theory.

Hilbert space 141

Examples

Lebesgue spaces

Lebesgue spaces are function spaces associated to measure spaces (X, M, ji), where X is a set, M is a o-algebra of

subsets of X, and fj, is a countably additive measure on M. Let L (X,\i) be the space of those complex-valued
measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is
finite, i.e., for a function /in L (X,\i),

f\ dfi < oo,
x

and where functions are identified if and only if they differ only on a set of measure zero.

The inner product of functions /and g in L (X,\i) is then defined as

(f,9)= I f{t)g{t) dp{t).
Jx

For/ and g in L , this integral exists because of the Cauchy— Schwarz inequality, and defines an inner product on the

space. Equipped with this inner product, L is in fact complete. The Lebesgue integral is essential to ensure

T211
completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.

2 2

The Lebesgue spaces appear in many natural settings. The spaces L (R) and L ([0,1]) of square-integrable functions

with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to

define the Fourier transform and Fourier series. In other situations, the measure may be something other than the

ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all

measurable functions/on the interval [0,1] satisfying

\f(t)\ 2 w(t)dt < oo

is called the weighted L space L , 2 , ,,([0,1]), and w is called the weight function. The inner product is defined by

</,<?) = t f(t)g(t)w(t)dt.
Jo

The weighted space L ,, ,,([0,1]) is identical with the Hilbert space L ([0,1], (x) where the measure u, of a
Lebesgue-measurable set A is defined by

fi{A) = I w{t)dt.

J A

Weighted L spaces like this are frequently used to study orthogonal polynomials, because different families of
orthogonal polynomials are orthogonal with respect to different weighting functions.

Sobolev spaces

v s 2

Sobolev spaces, denoted by H L or W ' , are Hilbert spaces. These are a special kind of function space in which
differentiation may be performed, but which (unlike other Banach spaces such as the Holder spaces) support the

structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the

T221
theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of

• *■ [23]

variations.

For s a non-negative integer and Q. C R", the Sobolev space If\Q.) contains L functions whose weak derivatives of

2 s

order up to s are also L . The inner product in H (Q.) is

(/, 9)= f f(x)g(x) dx+ f Df- Dg(x) dx + ---+ f D s f(x) ■ D'g(x) dx
Jo Jn Jn

where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces
can also be defined when s is not an integer.

Hilbert space 142

Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert
space structure. If Q is a suitable domain, then one can define the Sobolev space ff s (£2) as the space of Bessel
potentials; roughly,

F(fi) = {(i-A)- s / 2 /l/ei 2 (fi)}.

Here A is the Laplacian and (1 -A) is understood in terms of the spectral mapping theorem. Apart from

providing a workable definition of Sobolev spaces for non-integer s, this definition also has particularly desirable

properties under the Fourier transform that make it ideal for the study of pseudodifferential operators. Using these

methods on a compact Riemannian manifold, one can obtain for instance the Hodge decomposition which is the

[251
basis of Hodge theory.

Spaces of holomorphic functions

Hardy spaces

The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis, whose elements are
certain holomorphic functions in a complex domain. Let U denote the unit disc in the complex plane. Then the

Hardy space H (U)is defined to be the space of holomorphic functions /on U such that the means

remain bounded for r < 1 . The norm on this Hardy space is defined by
||/|| 2 = limVAW).

Hardy spaces in the disc are related to Fourier series. A function/is in H (U) if and only if

f{z) = Y,* n z n

71=0

where

El |2
\a n \ < oo.

71=0

2 2

Thus H (U) consists of those functions which are L on the circle, and whose negative frequency Fourier coefficients
vanish.

Bergman spaces

The Bergman spaces are another family of Hilbert spaces of holomorphic functions. Let D be a bounded open set

2 h

in the complex plane (or a higher dimensional complex space) and let L ' (D) be the space of holomorphic functions

/in D that are also in L (D) in the sense that

/ \m\ 2 dMz)

■ID

< CO,

2 h 2

where the integral is taken with respect to the Lebesgue measure in D. Clearly L ' (D) is a subspace of L (D); in fact,
it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on
compact subsets K of D, that

BUp|/(z)|<C*||/|| ai
zEK

which in turn follows from Cauchy's integral formula. Thus convergence of a sequence of holomorphic functions in

L (D) implies also compact convergence, and so the limit function is also holomorphic. Another consequence of this

2 h

inequality is that the linear functional that evaluates a function /at a point of D is actually continuous on L ' (D).

2 h

The Riesz representation theorem implies that the evaluation functional can be represented as an element of L ' (D).

2 h

Thus, for every z G D, there is a function n € L ' (D) such that

Hilbert space

143

/(z)= //(CK(C)<MC)

■Id

r 2,h

for all/€ L ' (D). The integrand

K(C,z)=r?,(C)
is known as the Bergman kernel of D. This integral kernel satisfies a reproducing property

f{*)= f f{QK{C,z)MO-

A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along

with a kernel K(t,,z) that verifies a reproducing property analogous to this one. The Hardy space H (D) also admits a
reproducing kernel, known as the Szego kernel. Reproducing kernels are common in other areas of mathematics
as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of
square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the
mean value theorem for harmonic functions.

Applications

Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple
geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the
spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral
decomposition of a matrix, and this often plays a major role in applications of the theory to other areas of
mathematics and physics.

Sturm-Liouville theory

In the theory of ordinary differential equations, spectral methods on a

suitable Hilbert space are used to study the behavior of eigenvalues and

eigenfunctions of differential equations. For example, the

Sturm— Liouville problem arises in the study of the harmonics of waves

in a violin string or a drum, and is a central problem in ordinary

[291
differential equations. The problem is a differential equation of the

form

The overtones of a vibrating string. These are

eigenfunctions of an associated Sturm— Liouville

problem. The eigenvalues 1,1/2,1/3,... form the

(musical) harmonic series.

d
dx

p(x)

q(x)y = Xw(x)y

for an unknown function y on an interval [a,b], satisfying general homogeneous Robin boundary conditions
ay(a) + a'y'(a) =

\(3y(b) + P'y'(b) = 0.

The functions p, q, and w are given in advance, and the problem is to find the function y and constants X for which
the equation has a solution. The problem only has solutions for certain values of X, called eigenvalues of the system,

Hilbert space 144

and this is a consequence of the spectral theorem for compact operators applied to the integral operator defined by
the Green's function for the system. Furthermore, another consequence of this general result is that the eigenvalues X
of the system can be arranged in an increasing sequence tending to infinity.

Partial differential equations

1221
Hilbert spaces form a basic tool in the study of partial differential equations. For many classes of partial

differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a

weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions,

which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem the analytic problem of

finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary

data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems

is the Lax— Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite element method) for

numerical solution of partial differential equations.

A typical example is the Poisson equation -Am = g with Dirichlet boundary conditions in a bounded domain CI in R .
The weak formulation consists of finding a function u such that, for all continuously differentiable functions v in Q.
vanishing on the boundary:

/ Vu ■ Vw = / gv.
Ja Ja

This can be recast in terms of the Hilbert space H 1(d) consisting of functions u such that u, along with its weak

partial derivatives, are square integrable on O, and which vanish on the boundary. The question then reduces to
finding u in this space such that for all v in this space

a(u, v) = b(y)
where a is a continuous bilinear form, and b is a continuous linear functional, given respectively by

a(u, v) = I V« ■ Vv, b(v) = / gv.
Ja Ja

Since the Poisson equation is elliptic, it follows from Poincare's inequality that the bilinear form a is coercive. The
Lax-Milgram theorem then ensures the existence and uniqueness of solutions of this equation.

Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the
Lax-Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be
applied to parabolic partial differential equations and certain hyperbolic partial differential equations.

Ergodic theory

The field of ergodic theory is the study of the long-term behavior of

chaotic dynamical systems. The protypical case of a field to which

ergodic theory is applicable is that of thermodynamics in which,

although the microscopic state of a system is extremely

complicated — it is impossible to understand the ensemble of individual

collisions between particles of matter — the average behavior over

sufficiently long time intervals is tractable. The laws of The path of a billiard ball in the Bunimovich

thermodynamics are assertions about such average behavior. In stadium is described by an ergodic dynamical

particular, one formulation of the zeroth law of thermodynamics system.

asserts that over sufficiently long timescales, the only functionally

independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the

form of temperature.

Hilbert space

145

An ergodic dynamical system is one for which, apart from the energy — measured by the Hamiltonian — there are no
other functionally independent conserved quantities on the phase space. More explicitly, suppose that the energy E is
fixed, and let O be the subset of the phase space consisting of all states of energy E (an energy surface), and let T
denote the evolution operator on the phase space. The dynamical system is ergodic if there are no continuous
non-constant functions on £2 „ such that

f(T t w) = f(w)

for all w on Q and all time t. Liouville's theorem implies that there exists a measure ll on the energy surface that is

invariant under the time translation. As a result, time translation is a unitary transformation of the Hilbert space

L (0_li) consisting of square-integrable functions on the energy surface Q with respect to the inner product

E E

(f,9)

L 2 (fW)

/ fgdn.

[19],

The von Neumann mean ergodic theorem states the following:

• If U is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space H, and P is the
orthogonal projection onto the space of common fixed points of U , {xEH I U x = x for all t > 0}, then

lira

1 fT

dt.

For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic

[32] 2

theorem implies the following: for any function/G L (Q, ,u),

L 2 -limi / f(T t w)dt= [ f(y)dfi(y).

T^oc T J J np;

That is, the long time average of an observable /is equal to its expectation value over an energy surface.

Fourier analysis

One of the basic goals of Fourier analysis is to decompose a function
into a (possibly infinite) linear combination of given basis functions:
the associated Fourier series. The classical Fourier series associated to
a function /defined on the interval [0,1] is a series of the form

Superposition of sinusoidal wave basis functions
(bottom) to form a sawtooth wave (top)

Hilbert space

146

Spherical harmonics, an orthonormal basis for the

Hilbert space of square-integrable functions on

the sphere, shown graphed along the radial

direction

£ a *

JlirinQ

where

On= I 1 f{B)l
JO

-2Trind

dd.

The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The
basis functions are sine waves with wavelengths "kin («=integer) shorter than the wavelength X of the sawtooth itself
(except for «=1, the fundamental wave). All basis functions have nodes at the nodes of the sawtooth, but all but the
fundamental have additional nodes. The oscillation of the summed terms about the sawtooth is called the Gibbs
phenomenon.

A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the
function/. Hilbert space methods provide one possible answer to this question. The functions e (8) = e mn form

2 "

an orthogonal basis of the Hilbert space L ([0,1]). Consequently, any square-integrable function can be expressed as
a series

f( G ) = ^2 a Ti e n{9), a n = (f, e„

and, moreover, this series converges in the Hilbert space sense (that is, in the L mean).

The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis, and
every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The

coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the

T341
element of the space. The abstraction is especially useful when it is more natural to use different basis functions

for a space such as L ([0,1]). In many circumstances, it is desirable not to decompose a function into trigonometric

functions, but rather into orthogonal polynomials or wavelets for instance, and in higher dimensions into spherical

u ■ [36]

harmonics.

2 2

For instance, if e are any orthonormal basis functions of L [0,1], then a given function in L [0,1] can be

[37]

approximated as a finite linear combination

fi?) ~ Ufa) = oiei(x) + a 2 e 2 {x)

^vfivXp^}

The coefficients {a.} are selected to make the magnitude of the difference \\f - f II as small as possible.
Geometrically, the best approximation is the orthogonal projection of /onto the subspace consisting of all linear

HQl

combinations of the {e.}, and can be calculated by

,-i:^m^.

That this formula minimizes the difference I/-/ II is a consequence of Bessel's inequality and Parseval's formula.

Hilbert space

147

In various applications to physical problems, a function can be decomposed into physically meaningful

eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral

[391
study of functions, in reference to the spectrum of the differential operator. A concrete physical application

involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is

capable of producing, can one infer the shape of the drum itself? The mathematical formulation of this question

involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of

vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.

Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis
decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to
the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function
defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier
transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an
isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of
the Fourier transformation is a recurring theme in abstract harmonic analysis, as evidenced for instance by the
Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.

Quantum mechanics

The orbitals of an electron in a hydrogen atom are
eigenfunctions of the energy.

In the mathematically rigorous formulation of quantum mechanics,

T411 [421

developed by Paul Dirac and John von Neumann , the possible

states (more precisely, the pure states) of a quantum mechanical system

are represented by unit vectors (called state vectors) residing in a

complex separable Hilbert space, known as the state space, well

defined up to a complex number of norm 1 (the phase factor). In other

words, the possible states are points in the projectivization of a Hilbert

space, usually called the complex projective space. The exact nature of

this Hilbert space is dependent on the system; for example, the position

and momentum states for a single non-relativistic spin zero particle is

the space of all square-integrable functions, while the states for the

spin of a single proton are unit elements of the two-dimensional

complex Hilbert space of spinors. Each observable is represented by a

self-adjoint linear operator acting on the state space. Each eigenstate of

an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value

of the observable in that eigenstate.

The time evolution of a quantum state is described by the Schrodinger equation, in which the Hamiltonian, the
operator corresponding to the total energy of the system, generates time evolution.

The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal
measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a
particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and
final states. The possible results of a measurement are the eigenvalues of the operator — which explains the choice of
self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given
state can be found by computing the spectral decomposition of the corresponding operator.

For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or
mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general
quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner
that is described instead by a positive operator valued measure. Thus the structure both of the states and observables

Hilbert space

148

in the general theory is considerably more complicated than the idealization for pure states.

Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain
observables do not commute, and gives a specific form that the commutator must have.

Properties
Pythagorean identity

Two vectors u and v in a Hilbert space H are orthogonal when iu, v) = 0. The notation for this is u L v. More

generally, when S is a subset in H, the notation u J. S means that u is orthogonal to every element from S.
When it and v are orthogonal, one has

\\u + v\\ = (u + v, u + v) = (u, u) + 2 Re{i/, v) + (v, v) = \\u\\ + ||u|| .
By induction on n, this is extended to any family u ,...,u of n orthogonal vectors,

1 1 ill + r- u n \\ 2 = ||ui|| 2 + h \\u n \\ .

Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the
extension of the Pythagorean identity to series. A series 2 u of orthogonal vectors converges in H if and only if the
series of squares of norms converges, and

Ell 2 V II l|2

fc=0 fc=0

Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.

Parallelogram identity and polarization

By definition, every Hilbert space is also a Banach space. Furthermore,
in every Hilbert space the following parallelogram identity holds:

Geometrically, the parallelogram identity asserts

that AC 2 + BD 2 = 2(AB 2 + AD 2 ). In words, the

sum of the squares of the diagonals is twice the

sum of the squares of any two adjacent sides.

|« + t;|| 2 + \\u - v\\ 2 = 2(\\u\\ 2 + \\v\\ 2 ).

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is
uniquely determined by the norm by the polarization identity. For real Hilbert spaces, the polarization identity is

v\\>)

\u,v) = - l||u + i>|| — \\u-
For complex Hilbert spaces, it is

(u,v) = - (\\u + v\\ 2 — \\u
The parallelogram law implies that any Hilbert space is a uniformly convex Banach space

1 1 2 * II ' II 2

v\\ +i\\u + iv\\

l\\U

iv\\ 2 ) .

[44]

Hilbert space 149

Best approximation

If C is a non-empty closed convex subset of a Hilbert space H and x a point in H, there exists a unique point y € C

[451
which minimizes the distance between x and points in C,

y (E C, \\x — y\\ = dist(x, C) = min{||x — z\\ :z£ C}.
This is equivalent to saying that there is a point with minimal norm in the translated convex set D = C - x. The proof
consists in showing that every minimizing sequence (d ) C D is Cauchy (using the parallelogram identity) hence
converges (using completeness) to a point in D that has minimal norm. More generally, this holds in any uniformly
convex Banach space.

When this result is applied to a closed subspace F of H, it can be shown that the point y € F closest to x is
characterized by

yGF, x-y±F.
This point y is the orthogonal projection of x onto F, and the mapping P : x — > y is linear (see Orthogonal
complements and projections). This result is especially significant in applied mathematics, especially numerical
analysis, where it forms the basis of least squares methods.

In particular, when F is not equal to H, one can find a non-zero vector v orthogonal to F (select x not in F and v = x -
y). A very useful criterion is obtained by applying this observation to the closed subspace F generated by a subset S
ofH.

A subset S of H spans a dense vector subspace if (and only if) the vector is the sole vector v € H orthogonal
to S.

Duality

The dual space H is the space of all continuous linear functions from the space H into the base field. It carries a
natural norm, defined by

H^ll — SUp |(/3(x)|.

||x||=i,xeff
This norm satisfies the parallelogram law, and so the dual space is also an inner product space. The dual space is also
complete, and so it is a Hilbert space in its own right.

The Riesz representation theorem affords a convenient description of the dual. To every element u of H, there is a
unique element q> of H , defined by

if u {x) = (x,u).
The mapping u i— > if a is an antilinear mapping from H to H . The Riesz representation theorem states that this
mapping is an antilinear isomorphism. Thus to every element cp of the dual H there exists one and only one u in
H such that

(x,u

<p(x)

for all x € H. The inner product on the dual space H satisfies

The reversal of order on the right-hand side restores linearity in q> from the antilinearity of u .In the real case, the
antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally
isomorphic to their own duals.

The representing vector u is obtained in the following way. When qp * 0, the kernel F = ker qp is a closed vector
subspace of H, not equal to H, hence there exists a non-zero vector v orthogonal to F. The vector u is a suitable
scalar multiple Xv of v. The requirement that q>{v) = (v, u) yields

u = (v,v)~ <p(v) V.

Hilbert space 150

This correspondence q> <-> u is exploited by the bra-ket notation popular in physics. It is common in physics to
assume that the inner product, denoted by (x\y), is linear on the right,

(*\v) = (v, x )-

The result (x\y) can be seen as the action of the linear functional (xl (the bra) on the vector ly) (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the
completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be
identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert
space H is reflexive, meaning that the natural map from H into its double dual space is an isomorphism.

Weakly convergent sequences

In a Hilbert space H, a sequence {x } is weakly convergent to a vector x € H when

1im{x n ,v) = (x,v)

for every v € H.

For example, any orthonormal sequence [f } converges weakly to 0, as a consequence of Bessel's inequality. Every
weakly convergent sequence {x } is bounded, by the uniform boundedness principle.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's

[49]
theorem). This fact may be used to prove minimization results for continuous convex functionals, in the same

way that the Bolzano-Weierstrass theorem is used for continuous functions on R . Among several variants, one

simple statement is as follows:

Iff: H — » R is a convex continuous function such that/(x) tends to +°° when llxll tends to °°, then/ admits a
minimum at some point x G H.

This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations.
Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that
closed bounded convex subsets in a Hilbert space H are weakly compact, since H is reflexive. The existence of
weakly convergent subsequences is a special case of the Eberlein-Smulian theorem.

Banach space properties

Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a
continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it
sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear
function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also
continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach

spaces. The open mapping theorem is equivalent to the closed graph theorem, which asserts that a function from

[52]
one Banach space to another is continuous if and only if its graph is a closed set. In the case of Hilbert spaces, this

is basic in the study of unbounded operators (see closed operator).

The (geometrical) Hahn— Banach theorem asserts that a closed convex set can be separated from any point outside it

by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation

property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane

[531
perpendicular to the segment xy passing through its midpoint.

Hilbert space 151

Operators on Hilbert spaces
Bounded operators

The continuous linear operators A : H — > H from a Hilbert space H to a second Hilbert space H are bounded in
the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous.
The space of such bounded linear operators has a norm, the operator norm given by

\\A\\ =stip{||Ae|| : ||x|| < 1}.
The sum and the composite of two bounded linear operators is again bounded and linear. For y in H , the map that
sends x € H to <Ax, y> is linear and continuous, and according to the Riesz representation theorem can therefore be
represented in the form

(x,A*y} = (Ax,y)
for some vector A yinH. This defines another bounded linear operator A ; H ' — > H ' the adjoint of A. One can see
that A -A.

The set B(H) of all bounded linear operators on H, together with the addition and composition operations, the norm
and the adjoint operation, is a C -algebra, which is a type of operator algebra.

An element A of B(H) is called self-adjoint or Hermitian if A = A. If A is Hermitian and {Ax, x) > for every x,
then A is called non-negative, written A > 0; if equality holds only when x = 0, then A is called positive. The set of
self adjoint operators admits a partial order, in which A > B if A - B > 0. If A has the form B B for some B, then A is
non-negative; if B is invertible, then A is positive. A converse is also true in the sense that, for a non-negative
operator A, there exists a unique non-negative square root B such that

A = B 2 =B*B.

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that
are "real". An element A of B(H) is called normal if A A = A A . Normal operators decompose into the sum of a
self-adjoint operators and an imaginary multiple of a self adjoint operator

, A + A* (A- A*)

A = h i- -

2 2i

that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary
parts.

An element U ofB(H) is called unitary if U is invertible and its inverse is given by U . This can also be expressed by
requiring that U be onto and (Ux, Uy) = (x, y) for all x and y in H. The unitary operators form a group under
composition, which is the isometry group of H.

An element of B(H) is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator
T is compact if, for any bounded sequence {x }, the sequence [TxA has a convergent subsequence. Many integral
operators are compact, and in fact define a special class of operators known as Hilbert— Schmidt operators that are
especially important in the study of integral equations. Fredholm operators are those which differ from a compact
operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel
and cokernel. The index of a Fredholm operator Tis defined by

index T — dim ker T — dim coker T.
The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah— Singer index
theorem.

Hilbert space 152

Unbounded operators

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics.
An unbounded operator T on a Hilbert space H is defined to be a linear operator whose domain D{T) is a linear
subspace of H. Often the domain D{T) is a dense subspace of H, in which case T is known as a densely defined
operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded
operators. Self-adjoint unbounded operators play the role of the observables in the mathematical formulation of
quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space L (R) are:

• A suitable extension of the differential operator

(Af)(x)=i^-f(x),
where i is the imaginary unit and/is a differentiable function of compact support.

• The multiplication-by-x operator:

{Bf){x) = xf{x).

These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on
all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be

square integrable. In both cases, the set of possible arguments form dense subspaces of L (R).

Constructions
Direct sums

Two Hilbert spaces H and H can be combined into another Hilbert space, called the (orthogonal) direct sum,
and denoted

consisting of the set of all ordered pairs (x , x ) where x. G H., i = 1,2, and inner product defined by

((xi,Z2),{V1,V2))h 1 ®H2 = (xuVliH! + (z 2 ,Ite)H a -
More generally, if His a family of Hilbert spaces indexed by i G 7, then the direct sum of the H., denoted

®*

iei
consists of the set of all indexed families

x= (xi GHi\i€l) ^Y[Hi

iei
in the Cartesian product of the H. such that

Ell Il2
Fill < oo.

iei

The inner product is defined by

iei
Each of the H. is included as a closed subspace in the direct sum of all of the H.. Moreover, the H. are pairwise

orthogonal. Conversely, if there is a system of closed subspaces V., i G I, in a Hilbert space H which are pairwise

orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of V.. In this case, H is

called the internal direct sum of the V.. A direct sum (internal or external) is also equipped with a family of

orthogonal projections E. onto the fth direct summand H.. These projections are bounded, self-adjoint, idempotent

operators which satisfy the orthogonality condition

Hilbert space 153

EiEj = 0, i± j.

The spectral theorem for compact self-adjoint operators on a Hilbert space H states that H splits into an orthogonal
direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of
projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock
space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds
to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter-Weyl
theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of
finite-dimensional representations.

Tensor products

If H and H , then one defines an inner product on the (ordinary) tensor product as follows. On simple tensors, let

(x 1 ®x 2y y! ®y 2 ) = {xuVi) {^2,2/2)-

This formula then extends by sesquilinearity to an inner product on H\ ® H 2 - The Hilbertian tensor product of H

and H , sometimes denoted by Hi®H 2 > * s tne Hilbert space obtained by completing H\ ® H 2 for me me tric

T571
associated to this inner product.

2 2

An example is provided by the Hilbert space L ([0, 1]). The Hilbertian tensor product of two copies of L ([0, 1]) is

2 2 2

isometrically and linearly isomorphic to the space L ([0, 1] ) of square-integrable functions on the square [0, 1] .
This isomorphism sends a simple tensor f± (x) ^to the function

(s,t) ^ Ms) f 2 (t)
on the square.

rcoi

This example is typical in the following sense. Associated to every simple tensor product X\ ® X2^ the rank
one operator

x* £ H* —s- x*[xi) x 2
from the (continuous) dual H to H . This mapping defined on simple tensors extends to a linear identification
between H\ ® H 2 ^nd the space of finite rank operators from H to H . This extends to a linear isometry of the
Hilbertian tensor product Hi®H 2 w ^ tne Hilbert space HS(H , H ) of Hilbert-Schmidt operators from H to

Orthonormal bases

[59]

The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert
space H, an orthonormal basis is a family {e } of elements of H satisfying the conditions:

1 . Orthogonality: Every two different elements of B are orthogonal: (e , e .)= for all k, j in B with k * j.

'- k j

2. Normalization: Every element of the family has norm V.We ,11 = 1 for all k in B.

3. Completeness: The linear span of the family e , k G B, is dense in H.

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set
(or an orthonormal sequence if B is countable). Such a system is always linearly independent. Completeness of an
orthonormal system of vectors of a Hilbert space can be equivalently restated as:

if (v, e j) = for all k € B and some v € H then v = 0.

This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if S is any
orthonormal set and v is orthogonal to S, then v is orthogonal to the closure of the linear span of S, which is the
whole space.

Examples of orthonormal bases include:

3
• the set {(1,0,0), (0,1,0), (0,0,1)} forms an orthonormal basis of R with the dot product;

Hilbert space 154

• the sequence {/ : n G Z} with/ (x) = exp(2jt/nx) forms an orthonormal basis of the complex space L ([0,1]);

In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish
the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense implies that every
vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this
decomposition is unique.

Sequence spaces

The space D of square-summable sequences of complex numbers is the set of infinite sequences

{ci,c 2 ,c 3 ,...)
of complex numbers such that

I 1 9 I 1 9 I |9

I c l| + \°i\ + M H < °°.

This space has an orthonormal basis:

e 1 = (1,0,0,...)
e 2 = (0,1,0,...)

More generally, if B is any set, then one can form a Hilbert space of sequences with index set B, defined by

£ 2 (B) = {x : B^C | J] \x(b)\ 2 < oo}.

beB
The summation over B is here defined by

J2 \ x ( b )\ 2 = su pE l x (MI 2

beB n=l

the supremum being taken over all finite subsets of B. It follows that, in order for this sum to be finite, every element

of D (B) has only countably many nonzero terms. This space becomes a Hilbert space with the inner product

( x >y) = J2 x ( b )y( b )

beB

for all x and y in D (B). Here the sum also has only countably many nonzero terms, and is unconditionally
convergent by the Cauchy— Schwarz inequality.

An orthonormal basis of D (B) is indexed by the set B, given by

e6(fc , )= jl \ib = b>

I otherwise.

Bessel's inequality and Parseval's formula

Let/ ,...,/ be a finite orthonormal system in H. For an arbitrary vector x in H, let

y = J2( x ifj)fj-

Then (x,/) = (y,/) for every k- 1, ..., n. It follows thatx - y is orthogonal to each/ , hence x - y is orthogonal to y.
Using the Pythagorean identity twice, it follows that

n
II 1 1 2 II ||2 i || ||2 ^ || ||2 V^ | / r \ |2

Nl =\\ x -y\\ +\\y\\ >\\y\\ =2Jv c >/>)l ■

3=1

Let {/ }, i € I, be an arbitrary orthonormal system in H. Applying the preceding inequality to every finite subset J of
/ gives the Bessel inequality

Hilbert space 155

iei
(according to the definition of the sum of an arbitrary family of non-negative real numbers).

Geometrically, Bessel's inequality implies that the orthogonal projection of x onto the linear subspace spanned by the
/. has norm that does not exceed that of x. In two dimensions, this is the assertion that the length of the leg of a right
triangle may not exceed the length of the hypotenuse.

Bessel's inequality is a stepping stone to the more powerful Parseval identity which governs the case when Bessel's
inequality is actually an equality. If {e } is an orthonormal basis of H, then every element x of H may be written

k k G B

x = X ( x > efe ) efe -
keB

Even if B is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of
countably many nonzero terms. This sum is called the Fourier expansion of x, and the individual coefficients (;t,e,)
are the Fourier coefficients of x. Parseval's formula is then

||z|| 2 = £!<*, e fc )| 2 .

keB

ersely, if {

basis

Conversely, if {e } is an orthonormal set such that Parseval's identity holds for every x, then {e } is an orthonormal

Hilbert dimension

As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two
orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space. For
instance, since D (B) has an orthonormal basis indexed by B, its Hilbert dimension is the cardinality of B (which may
be a finite integer, or a countable or uncountable cardinal number).

As a consequence of Parseval's identity, if {e } is an orthonormal basis of H, then the map <J> : H — > I (B)

k k G B

defined by O(x) = ((x,e )) is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that

kf kGd

<$0),$(y))^(B) = (x,y) H

for all x and y in H. The cardinal number of B is the Hilbert dimension of H. Thus every Hilbert space is

isometrically isomorphic to a sequence space £ (B) for some set B.

Separable spaces

A Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable
Hilbert spaces are therefore isometrically isomorphic to £ 2 .

In the past, Hilbert spaces were often required to be separable as part of the definition. Most spaces used in
physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional
separable Hilbert space as "the Hilbert space" or just "Hilbert space". Even in quantum field theory, most of the
Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that
non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory
possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension
greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a
tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural
state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of
the full tensor product that can contain physically meaningful fields (on which the observables can be defined).
Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of
space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since

Hilbert space 156

the set of possible densities is uncountable, the basis is not countable.

Orthogonal complements and projections

If S is a subset of a Hilbert space H, the set of vectors orthogonal to S is defined by

S ± = {x£H : (x,s) = 0\/s€S}.
S is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then V is called the
orthogonal complement of V. In fact, every x in H can then be written uniquely as x = v + w, with v in V and w in V .
Therefore, H is the internal Hilbert direct sum of V and v.

The linear operator P : H — > H which maps x to v is called the orthogonal projection onto V. There is a natural
one-to-one correspondence between the set of all closed subspaces of H and the set of all bounded self-adjoint

operators P such that P = P. Specifically,

Theorem. The orthogonal projection P is a self-adjoint linear operator on H of norm < 1 with the property

2 2

P — P . Moreover, any self-adjoint linear operator E such that E = E is of the form P p where V is the range
of E. For every x in H, P v ( x ) is the unique element v of V which minimizes the distance I be - vll.
This provides the geometrical interpretation of P v (x): it is the best approximation to x by elements of V.

2 *

An operator P such that P = P = P is called an orthogonal projection. The orthogonal projection P onto a closed
subspace Vof H is the adjoint of the inclusion mapping

i v : V -> H,
meaning that

(i v x, y) = {x, P v y)
for all x € H and y G V. Projections P and ^* v are called mutually orthogonal if P,jP v = 0. This is equivalent to U
and V being orthogonal as subspaces of H. As a result, the sum of the two projections P and P,A S on ly a projection
if U and V are orthogonal to each other, and in that case P + P - P . The composite P P is generally not a
projection; in fact, the composite is a projection if and only if the two projections commute, and in that case

P P =P

IT V UnV

The operator norm of a projection P onto a non-zero closed subspace is equal to one:

||Fx||

||P|| = SUp — — :— = 1.

xeH,x=£G \\x\\

Every closed subspace V of a Hilbert space is therefore the image of an operator P of norm one such that P = P. In
fact this property characterizes Hilbert spaces:

• A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, to every closed
subspace V, there is an operator P„of norm one whose image is V such that p"K = P v .

While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a topological
vector space can itself be characterized in terms of the presence of complementary subspaces:

• A Banach space X is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed
subspace V, there is a closed subspace W such that X is equal to the internal direct sum V © W ■

The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if
£/" C V> m en y-^- C U^ vvith equality holding if and only if V is contained in the closure of U. This result is a
special case of the Hahn— Banach theorem. The closure of a subspace can be completely characterized in terms of the
orthogonal complement: If V is a subspace of H, then the closure of V is equal to \Z- L - L • The orthogonal
complement is thus a Galois connection on the partial order of subspaces of a Hilbert space. In general, the
orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements:
(Ei V i ) ± = Hi Vt ■ If the V i are in addition closed, then Y~V± = (f\ V i ) ± ■

Hilbert space 157

Spectral theory

There is a well-developed spectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to
the study of symmetric matrices over the reals or self-adjoint matrices over the complex numbers. In the same
sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of
orthogonal projection operators.

The spectrum of an operator T, denoted o(7) is the set of complex numbers X such that T - X lacks a continuous
inverse. If T is bounded, then the spectrum is always a compact set in the complex plane, and lies inside the disc
|a|<||T||. If Tis self-adjoint, then the spectrum is real. In fact, it is contained in the interval \m,M\ where

77i— inf (Tx,x), M= sup (Tx, x).

Nl =1 ||x||=i

Moreover, m and M are both actually contained within the spectrum.
The eigenspaces of an operator T are given by

fl' A = ker(r-A).

Unlike with finite matrices, not every element of the spectrum of T must be an eigenvalue: the linear operator T -X
may only lack an inverse because it is not surjective. Elements of the spectrum of an operator in the general sense are
known as spectral values. Since spectral values need not be eigenvalues, the spectral decomposition is often more
subtle than in finite dimensions.

However, the spectral theorem of a self-adjoint operator T takes a particularly simple form if, in addition, T is
assumed to be a compact operator. The spectral theorem for compact self-adjoint operators states:

• A compact self-adjoint operator 7" has only countably (or finitely) many spectral values. The spectrum of T has no
limit point in the complex plane except possibly zero. The eigenspaces of T decompose H into an orthogonal
direct sum:

Aecr(T)

Moreover, if E denotes the orthogonal projection onto the eigenspace H , then

A. A.

Ae<r(T)

where the sum converges with respect to the norm on B(H).

This theorem plays a fundamental role in the theory of integral equations, as many integral operators are compact, in
particular those that arise from Hilbert-Schmidt operators.

The general spectral theorem for self-adjoint operators involves a kind of operator-valued Riemann— Stieltjes
integral, rather than an infinite summation. The spectral family associated to T associates to each real number X an
operator E , which is the projection onto the nullspace of the operator (T — A) + » where the positive part of a

self-adjoint operator is defined by

The operators E are monotone increasing relative to the partial order defined on self-adjoint operators; the

5 1

eigenvalues correspond precisely to the jump discontinuities. One has the spectral theorem, which asserts

T= I XdE x .

it
The integral is understood as a Riemann— Stieltjes integral, convergent with respect to the norm on B(H). In

particular, one has the ordinary scalar-valued integral representation

(Tx,y) = / Xd(E x x,y).

Hilbert space 158

A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now
contain non-real complex numbers, the operator-valued Stieltjes measure dE must instead be replaced by a

resolution of the identity.

A major application of spectral methods is the spectral mapping theorem, which allows one to apply to a self-adjoint
operator T any continuous complex function /defined on the spectrum of Tby forming the integral

f(T)= f f(X)dE,.

Jcr(T)

r(T)

T721
The resulting continuous functional calculus has applications in particular to pseudodifferential operators.

The spectral theory of unbounded self-adjoint operators is only marginally more difficult than for bounded operators.
The spectrum of an unbounded operator is defined in precisely the same way as for bounded operators: X is a spectral
value if the resolvent operator

R X = {T- A)- 1

fails to be a well-defined continuous operator. The self-adjointness of T still guarantees that the spectrum is real.
Thus the essential idea of working with unbounded operators is to look instead at the resolvent R where X is

non-real. This is a bounded normal operator, which admits a spectral representation that can then be transferred to a
spectral representation of T itself. A similar strategy is used, for instance, to study the spectrum of the Laplace
operator: rather than address the operator directly, one instead looks as an associated resolvent such as a Riesz
potential or Bessel potential.

T731
A precise version of the spectral theorem which holds in this case is:

Given a densely defined self-adjoint operator T on a Hilbert space H, there corresponds a unique resolution of
the identity E on the Borel sets of R, such that

(Tx,y)= f XdE x , y (X)

for all x € D{T) and y € H. The spectral measure E is concentrated on the spectrum of T.
There is also a version of the spectral theorem that applies to unbounded normal operators.

Notes

[I] Marsden 1974, §2.8

[2] The mathematical material in this section can be found in any good textbook on functional analysis, such as Dieudonne (1960), Hewitt &

Stromberg (1965), Reed & Simon (1980) or Rudin (1980).
[3] In some conventions, inner products are linear in their second arguments instead.
[4] Dieudonne 1960, §6.2
[5] Dieudonne 1960
[6] Largely from the work of Hermann Grassmann, at the urging of August Ferdinand Mobius (Boyer & Merzbach 1991, pp. 584—586). The first

modern axiomatic account of abstract vector spaces ultimately appeared in Giuseppe Peano's 1888 account (Grattan-Guinness 2000, §5.2.2;

O'Connor & Robertson 1996).
[7] A detailed account of the history of Hilbert spaces can be found in Bourbaki 1987.
[8] Schmidt 1908
[9] Titchmarsh 1946, §IX.l
[10] Lebesgue 1904. Further details on the history of integration theory can be found in Bourbaki (1987) and Saks (2005).

[II] Bourbaki 1987.

[12] Dunford & Schwartz 1958, §IV.16

[13] In Dunford & Schwartz (1958, §IV.16), the result that every linear functional on L [0,1] is represented by integration is jointly attributed to

Frechet (1907) and Riesz (1907). The general result, that the dual of a Hilbert space is identified with the Hilbert space itself, can be found in

Riesz (1934).
[14] von Neumann 1929.
[15] Kline 1972, p. 1092
[16] Hilbert, Nordheim & von Neumann 1927.
[17] Weyl 1931.

Hilbert space

159

[18
[19

[20
[21
[22
[23
[24
[25
[26
[27
[28
[29
[30
[31
[32
[33
[34
[35
[36
[37
[38
[39
[40
[41
[42
[43
[44
[45
[46
[47
[48
[49
[50
[51
[52
[53
[54
[55
[56
[57
[58
[59
[60
[61

[62
[63

[64
[65
[66
[67
[68
[69

[70

[71

[72

Prugovecki 1981, pp. 1-10.

von Neumann 1932

Halmos 1957, Section 42.

Hewitt & Stromberg 1965.

Bers, John & Schechter 1981.

Giusti 2003.

Stein 1970

Details can be found in Warner (1983).

A general reference on Hardy spaces is the book Duren (1970).

Krantz 2002, §1.4

Krantz 2002, §1.5

Young 1988, Chapter 9.

The eigenvalues of the Fredholm kernel are 1/)., which tend to zero.

More detail on finite element methods from this point of view can be found in Brenner & Scott (2005).

Reed & Simon 1980

A treatment of Fourier series from this point of view is available, for instance, in Rudin (1987) or Folland (2009).

Halmos 1957, §5

Bachman, Narici & Beckenstein 2000

Stein & Weiss 1971, §IV.2.

Lancos 1988, pp. 212-213

Lanczos 1988, Equation 4-3.10

The classic reference for spectral methods is Courant & Hilbert 1953. A more up-to-date account is Reed & Simon 1975.

Kac 1966

Dirac 1930

von Neumann 1955

Young 1988, p. 23.

Clarkson 1936.

Rudin 1987, Theorem 4.10

Dunford & Schwartz 1958, II.4.29

Rudin 1987, Theorem 4.1 1

Weidmann 1980, Theorem 4.8

Weidmann 1980, §4.5

Buttazzo, Giaquinta & Hildebrandt 1998, Theorem 5.17

Halmos 1982, Problem 52, 58

Rudin 1973

Treves 1967, Chapter 18

See Prugovecki (1981), Reed & Simon (1980, Chapter VIII) and Folland (1989).

Prugovecki 1981, III, §1.4

Dunford & Schwartz 1958, IV.4.17-18

Weidmann 1980, §3.4

Kadison & Ringrose 1983, Theorem 2.6.4

Dunford & Schwartz 1958, §IV.4.

For the case of finite index sets, see, for instance, Halmos 1957, §5. For infinite index sets, see Weidmann 1980, Theorem 3.6.

Levitan 2001. Many authors, such as Dunford & Schwartz (1958, §IV.4), refer to this just as the dimension. Unless the Hilbert space is finite

dimensional, this is not the same thing as its dimension as a linear space (the cardinality of a Hamel basis).

Prugovecki 1981,1, §4.2

von Neumann (1955) defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with £^. The

convention still persists in most rigorous treatments of quantum mechanics; see for instance Sobrino 1996, Appendix B.
Streater & Wightman 1964, pp. 86-87

Young 1988, Theorem 15.3

Kakutani 1939

Lindenstrauss & Tzafriri 1971

Halmos 1957, §12

A general account of spectral theory in Hilbert spaces can be found in Riesz & Sz Nagy (1990). A more sophisticated account in the
anguage of C -algebras is in Rudin (1973) or Kadison & Ringrose (1997)

See, for instance, Riesz & Sz Nagy (1990, Chapter VI) or Weidmann 1980, Chapter 7. This result was already known to Schmidt (1907) in
the case of operators arising from integral kernels.

Riesz & Sz Nagy 1990, §§107-108

Shubin 1987

Hilbert space 160

[73] Rudin 1973, Theorem 13.30.

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Streater, Ray; Wightman, Arthur (1964), PCT, Spin and Statistics and All That, W. A. Benjamin, Inc.

Hilbert space

162

• Titchmarsh, Edward Charles (1946), Eigenfunction expansions, part 1, Oxford University: Clarendon Press.

• Treves, Francois (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press.

• von Neumann, John (1929), "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren", Mathematische
Annalen 102: 49-131, doi:10.1007/BF01782338.

• von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA 18 (3):
263-266, doi:10.1073/pnas.l8.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.

• von Neumann, John (1955), Mathematical foundations of quantum mechanics, Princeton Landmarks in
Mathematics, Princeton University Press (published 1996), ISBN 978-0-691-02893-4, MR1435976.

• Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York:
Springer- Verlag, ISBN 978-0-387-90894-6.

• Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, 68, Berlin, New
York: Springer- Verlag, ISBN 978-0-387-90427-6, MR566954.

• Weyl, Hermann (1931), The Theory of Groups and Quantum Mechanics (English 1950 ed.), Dover Press,
ISBN 0-486-60269-9.

• Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 0-521-33071-8,
Zbl: 0645.46024.

External links

• Hilbert Space at Mathworld (http://mathworld.wolfram.com/HilbertSpace.html)

• 245B, notes 5: Hilbert spaces (http://terrytao.wordpress.com/2009/01/17/254a-notes-5-hilbert-spaces/) by
Terence Tao

Spherical harmonics

In mathematics, spherical harmonics are

the angular portion of a set of solutions to
Laplace's equation. Represented in a system
of spherical coordinates, Laplace's spherical
harmonics l^are a specific set of
spherical harmonics that forms an
orthogonal system, first introduced by Pierre
Simon de Laplace. Spherical harmonics
are important in many theoretical and
practical applications, particularly in the
computation of atomic orbital electron
configurations, representation of

gravitational fields, geoids, and the

*******

Visual representations of the first few spherical harmonics. Red portions represent

regions where the function is positive, and green portions represent regions where

the function is negative.

magnetic fields of planetary bodies and

stars, and characterization of the cosmic microwave background radiation. In 3D computer graphics, spherical
harmonics play a special role in a wide variety of topics including indirect lighting (ambient occlusion, global
illumination, precomputed radiance transfer, etc.) and recognition of 3D shapes.

Spherical harmonics 163

History

Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal
gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mecanique Celeste, determined that the
gravitational potential at a point x associated to a set of point masses m . located at points x . was given by
m;

vw = £

Xj -x

■I

Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time,
Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = Ixl and r = Ix I.
He discovered that if r < r then

1 1 r r 2

-. r = P (COS7) h P^COS'y)-^ + P 2 (cOS7)^ -\

|xi — x| T\ r{ rf

where y is the angle between the vectors x and x . The functions P. are the Legendre polynomials, and they are a
special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coefficients
using spherical coordinates to represent the angle y between x and x. (See Applications of Legendre polynomials in
physics for a more detailed analysis.)

In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their
Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. The
solid harmonics were homogeneous solutions of Laplace's equation

d 2 u d 2 u d 2 u
dx 2 dy 2 dz 2

By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical
harmonics. The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of
solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics
that had properly been introduced by Laplace and Legendre.

The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in
rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by
expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series
represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes
of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by
taking expansions in spherical harmonics rather than trigonometric functions. This was a boon for problems
possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.

The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century
birth of quantum mechanics. The spherical harmonics are eigenfunctions of the square of the orbital angular
momentum operator

-ikr X V,
and therefore they represent the different quantized configurations of atomic orbitals.

Spherical harmonics

164

Laplace's spherical harmonics

Laplace's equation imposes that the divergence of the gradient of a scalar field
/is zero. In spherical coordinates this is:

Real (Laplace) spherical harmonics
Y™for£ = 0to4(topto

bottom) and fTl = to 4(' e f"
right). The negative order harmonics
y„ — m are rotated about the 5? axis

by 90 /?7l with respect to the
positive order ones.

* r 2 dr \ 8r) + r 2 wa.6 88 \ 89

8 2 f

r 2 sin dip 2

Consider the problem of finding solutions of the form /(r,6,cp) = /?(r)F(0,cp), By separation of variables, two
differential equations result by imposing Laplace's equation:

1 1 8 ( . 8Y\ 1 1 8 2 Y

= A, sin0 -\ ^ -

--■-"-' Qd ) Y sin 2 6 dip 2

= -A.

dr J ' Y sin 9 88

The second equation can be simplified under the assumption that Y has the form Y(Q,(p) = ©(6)0(ep). Applying
separation of variables again to the second equation gives way to the pair of differential equations

1 d?®{ip)

$(cp) dip 2
Asin 2 (0) +

-m

sin

(9) d

6(0) d6

for some number m. A priori, m is a complex constant, but because O must be a periodic function whose period
evenly divides 2n, m is necessarily an integer and O is a linear combination of the complex exponentials <?~' mcf '. The
solution function F(8,cp) is regular at the poles of the sphere, where 6=0,Jt. Imposing this regularity in the solution ©
of the second equation at the boundary points of the domain is a Sturm— Liouville problem that forces the parameter
X to be of the form X = £{£+!) for some non-negative integer with £ > \m\; this is also explained below in terms of the
orbital angular momentum. Furthermore, a change of variables t = cos6 transforms this equation into the Legendre
equation, whose solution is a multiple of the associated Legendre polynomial P^f cos 6) ■ Finally, the equation for
R has solutions of the form R(r) = Ar + Br ; requiring the solution to be regular throughout R forces B = 0.
Here the solution was assumed to have the special form y(6,cp) = 0(6)O(cp). For a given value of t, there are 2£+\
independent solutions of this form, one for each integer m with -£<m<{. These angular solutions are a product of
trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

Y™(6, <f)=N e im(fi P™(cos 9),
which fulfill

r 2 V 2 Y g m (9 : ip) = -£(£ + i)YT(6, if).

Spherical harmonics 165

Here Y™is called a spherical harmonic function of degree I and order m, /'/"is an associated Legendre
polynomial, A' is a normalization constant, and 6 and cp represent colatitude and longitude, respectively. In particular,
the colatitude 6, or polar angle, ranges from at the North Pole to Jt at the South Pole, assuming the value of jc/2 at
the Equator, and the longitude cp, or azimuth, may assume all values with < cp < 2jt. For a fixed integer (, every
solution y(6,cp) of the eigenvalue problem

r 2 v 2 r = -£(e + i)Y

p
is a linear combination of Y™ . In fact, for any such solution, r y(6,cp) is the expression in spherical coordinates of

a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2(+\

linearly independent such polynomials.

The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical

harmonic functions multiplied by the appropriate scale factor r ,

£=0 m=-£

where the //"are constants and the factors ^ Y7"are known as solid harmonics. Such an expansion is valid in the
ball

r <R = l/limsupl/7 1 ! 1 ^.

£->ac

Orbital angular momentum

mi
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum

L = —ihx. x V = L x \ + L y ] + L z k.

The /j, is conventional in quantum mechanics; it is convenient to work in units in which /j, = \. The spherical
harmonics are eigenfunctions of the square of the orbital angular momentum

.a id 1

sino-

sin 9 36 86 sin 2 9 dip 2 '
Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the
generator of rotations about the azimuthal axis:

8 8

x y —

dy ox

8(f

These operators commute, and are densely defined self-adjoint operators on the Hilbert space of functions /

square-integrable with respect to the normal distribution on R :

(2t)

\^ [ \f(x)\ 2 e-^ 2 dx<oc.

Furthermore, L is a positive operator.

If Y is a joint eigenfunction of L and L , then by definition

L 2 y = XY

L Z Y = mY
for some real numbers m and X. Here m must in fact be an integer, for Y must be periodic in the coordinate cp with
period a number that evenly divides 2jc. Furthermore, since

Spherical harmonics 166

L 2 = L 2 + L 2 + L 2

and each of L , L , L are self-adjoint, it follows that "k>m.

x y z

Denote this joint eigenspace by E , and define the raising and lowering operators by
L_|_ = L x + iL y

L- = L x — iL y

Then L and L commute with L , and the Lie algebra generated by L , L , L is the special linear Lie algebra, with
commutation relations

[L a , L+] = L + , [L a , LJ\ = -L_, [L+, L_] = 2L Z .

Thus L : E. — > E. , (it is a "raising operator") and L : E. — > £, , (it is a "lowering operator"). In particular,
+ X,»i X,m+1 ° r X,m X,m-1 9

L fe : £, — » £, , must be zero for fc sufficiently large, because the inequality X > m must hold in each of the

+ X,m X,m+k jo' i j

nontrivial joint eigenspaces. Let Y£ E be a nonzero joint eigenfunction, and let k be the least integer such that

L\Y = 0.
Then, since

L-L + = L — L z — L z

it follows that

= L_L\Y = (A - (m + kf - (m + fc))y.
Thus X = £(£+1) for the positive integer £ = m+fe.

Conventions

Orthogonality and normalization

Several different normalizations are in common use for the Laplace spherical harmonic functions. Throughout the
section, we use the standard convention that (see associated Legendre polynomials)

(£ — mV
(£ + m) !
which is the natural normalization given by Rodrigues' formula.

In physics and seismology, the Laplace spherical harmonics are generally defined as

YTV> v)

(2£ + l)(£-m)\ imip

which are orthonormal

Jg=o Jtp=o

Y™ Y™ * d£l — Si? 8 mm r ,

where 5 =1,5=0 if a # b, (see Kronecker delta) and dQ = sin6 dtp dQ. This normalization is used in quantum

£13. 3.D f,

mechanics because it ensures that probability is normalized, i.e. / \YZ n \ (Kl = 1 • The disciplines of geodesy and
spectral analysis use

which possess unit power

^ + 1 >§T^T p < m < c ° s ^'" , '

2 1"K /*2tt
4lT J9=0 Jip=0

Y e m Y £ T r*<m = 5 MI 5 T ,

Spherical harmonics 167

The magnetics community, in contrast, uses Schmidt semi-normalized harmonics

Yr(o, & =

which have the normalization

^ m)! P™(co S d)e im *

fTT flit

J6=0 J<p=0

2n YrY e r'*dn = j^ T) 5wS T ,

In quantum mechanics this normalization is often used as well, and is named Racah's normalization after Giulio
Racah.

It can be shown that all of the above normalized spherical harmonic functions satisfy

Yr{e,<p) = (-i) m Yr n (o,< P ),

where the superscript * denotes complex conjugation. Alternatively, this equation follows from the relation of the
spherical harmonic functions with the Wigner D-matrix.

Condon-Shortley phase

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of (-1) m ,
commonly referred to as the Condon— Shortley phase in the quantum mechanical literature. In the quantum
mechanics community, it is common practice to either include this phase factor in the definition of the associated
Legendre polynomials, or to append it to the definition of the spherical harmonic functions. There is no requirement
to use the Condon— Shortley phase in the definition of the spherical harmonic functions, but including it can simplify
some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy and
magnetics communities never include the Condon— Shortley phase factor in their definitions of the spherical
harmonic functions.

Real form

A real basis of spherical harmonics can be defined in terms of their complex analogues by setting

f 72 ( y " + (- 1 )™ Y t m ) = V2N {e , m) Pp(coa 6) cos rmp if m >
Y lm = I Y e ° if m =

{-^{yf m - (-i) m >r) = v^% m )ir"(cos0)siiim<p if m < o.

where Nu m \ denotes the normalization constant as a function of I and m ■ The real form requires only associated
Legendre polynomials p\ Tn \ of non-negative \m\. The harmonics with m > are said to be of cosine type, and those

with m < of sine type. These real spherical harmonics are sometimes known as tesseral spherical harmonics.
These functions have the same normalization properties as the complex ones above. See here for a list of real
spherical harmonics up to and including £ = 5- Note, however, that the listed functions differ by the phase (-1)
from the phase given in this article.

Spherical harmonics 168

Spherical harmonics expansion

The Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis
of the Hilbert space of square-integrable functions. On the unit sphere, any square-integrable function can thus be
expanded as a linear combination of these:
oo (.

Mv) = £ E fTYTfrv)-

£=0m=-e

This expansion holds in the sense of mean-square convergence — convergence in L of the sphere — which is to say
that

2 N £

lim f r f (9^)-^ E fTYT{0,<P) sm0d0d0 = O.

The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above
equation by the complex conjugate of a spherical harmonic, integrating over the solid angle fl, and utilizing the
above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of
orthonormalized harmonics, this gives:

r /*2tt riv

f? = / f(e, ip) ytV, ip)dn= d^ dd ^e f (e, tfvrie, ?).

Jn Jo Jo

If the coefficients decay in I sufficiently rapidly — for instance, exponentially — then the series also converges
uniformly to /

A real square-integrable function/can be expanded in terms of the real harmonics Y above as a sum
oc (.

/(*,¥>) = £ E h m YU$M-

£=0 m=-£

Convergence of the series holds again in the same sense.

Spectrum analysis

Power spectrum in signal processing

The total power of a function / is defined in the signal processing literature as the integral of the function squared,
divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonic
functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its
spectral coefficients by a generalization of Parseval's theorem:

1 r °°

— /(n) a dn = £-%(*),

4 7T JO. g =0

where

SfAi) = E JL

m=-£

is defined as the angular power spectrum. In a similar manner, one can define the cross-power of two functions as

If °°

— f(n) g {n)dsi = Y,s fa {i),

4 7r Jn i= Q

where

^/fl(^) = E fe™g£m

Tn,=—£

Spherical harmonics 169

is defined as the cross-power spectrum. If the functions / and g have a zero mean (i.e., the spectral coefficients/
and g are zero), then S A€) and S {€) represent the contributions to the function's variance and covariance for
degree I, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the
form

S ff (£) = C£f>.
When p = 0, the spectrum is "white" as each degree possesses equal power. When |3 < 0, the spectrum is termed
"red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when p > 0, the
spectrum is termed "blue". The condition on the order of growth of S Al) is related to the order of differentiability of
/in the next section.

Differentiability properties

One can also understand the differentiability properties of the original function/in terms of the asymptotics of S Al).
In particular, if S Al) decays faster than any rational function of I as I — > °°, then / is infinitely differentiable. If,
furthermore, S At) decays exponentially, then/is actually real analytic on the sphere.

The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the S Al) to
differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically,
if

Y,(i+e 2 ys ff (£)<^

s 2

then / is in the Sobolev space H' (S ). In particular, the Sobolev embedding theorem implies that / is infinitely
differentiable provided that

S n {l) = 0(r s ) as I -> oo
for all s.

Algebraic properties
Addition theorem

A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. This is
a generalization of the trigonometric identity

cos(#' — 9) — cos 9' cos 9 + sin 9 sin 6'
in which the role of the trigonometric functions appearing on the right-hand side is played by the spherical
harmonics and that of the left-hand side is played by the Legendre polynomials.

Consider two unit vectors x and y, having spherical coordinates (6,cp) and (6',q/), respectively. The addition theorem
states

Pfr ■ y) = 2^ E y ^'> vO Y ^ e > *)■ (1)

m=-£

where P is the Legendre polynomial of degree I. This expression is valid for both real and complex harmonics.
The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by
applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand
side. [7]

Spherical harmonics 170

ro]

In particular, when x = y, this gives Unsold's theorem

1 2^ + 1

£ YL(8, <p) YiJfii *0 = ~^~

m=—l

2 2

which generalizes the identity cos 6 + sin 6 = 1 to two dimensions.

In the expansion (1), the left-hand side P (x-y) is a constant multiple of the degree ( zonal spherical harmonic. From
this perspective, one has the following generalization to higher dimensions. Let Y. be an arbitrary orthonormal basis
of the space H of degree ( spherical harmonics on the n-sphere. Then Z^> the degree ( zonal harmonic

corresponding to the unit vector x, decomposes as

dim(H,)

(2)

Furthermore, the zonal harmonic 2^)(y)is given as a constant multiple of the appropriate Gegenbauer
polynomial: {{{}}}

^(y) = ^- 1)/2) (x-y)

(3)

{{{}}} Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates.
Finally, evaluating at x = y gives the functional identity

dim(H £ )

dimH < £ W( x)r-

J=l
where <x> is the volume of the («-l)-sphere.

Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical
harmonics in terms of spherical harmonics itself. A variety of techniques are available for doing essentially the same
calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. Abstractly, the
Clebsch-Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a
sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.

Parity

The spherical harmonics have well defined parity in the sense that they are either even or odd with respect to
reflection about the origin. Reflection about the origin is represented by the operator P^f(r) = ^f(-r)- For the
spherical angles, |0 ; (j)\ this corresponds to the replacement i% — 9, 7T + (j)\ ■ The associated Legendre
polynomials gives (-1) " and from the exponential function we have (-1) , giving together for the spherical
harmonics a parity of (-1) :

Y e m (9, (f>) -► Yr(7r -9,7: + ^) = (-lyrne, J))

This remains true for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic
of degree ( changes the sign by a factor of (-1) .

Spherical harmonics

171

Visualization of the spherical harmonics

The Laplace spherical harmonics YT^can
be visualized by considering their "nodal
lines", that is, the set of points on the sphere
where Yf 1 = 0. Nodal lines of Y™are
composed of circles: some are latitudes and
others are longitudes. One can determine the
number of nodal lines of each type by
counting the number of zeros of Y™in the
latitudinal and longitudinal directions
independently. For the latitudinal direction,
the associated Legendre polynomials
possess l-\m\ zeros, whereas for the
longitudinal direction, the trigonometric sin
and cos functions possess 21ml zeros.

When the spherical harmonic order m is
zero (upper-left in the figure), the spherical
harmonic functions do not depend upon
longitude, and are referred to as zonal. Such
spherical harmonics are a special case of
zonal spherical functions. When ( = \m\
(bottom-right in the figure), there are no
zero crossings in latitude, and the functions
are referred to as sectoral. For the other
cases, the functions checker the sphere, and
they are referred to as tesseral.

Schematic representation of Yg m on the unit sphere and its nodal lines. Yg m is

equal to along TYl great circles passing through the poles, and along £ — jji

circles of equal latitude. The function changes sign each time it crosses one of

these lines.

More general spherical harmonics of degree ( are not necessarily those of the Laplace basis Y™ , and their nodal
sets can be of a fairly general kind.

List of spherical harmonics

Analytic expressions for the first few orthonormalized Laplace spherical harmonics that use the Condon-Shortley
phase convention:

Spherical harmonics

172

n=5 m=0

LJj

[03

E3ta

L- 3f • 31

r«M>U<M»j|

3D color plot of the spherical harmonics of
degree n = 5

Y °(9,<p)

1 1

Y^{6^)

_ 1 /15

4V7T

-l fiE

~2~V 2^
1 /l5

sin# cos#e lv °
(3cos 2 0-l)
sin cos 6 e llp

Y W^ = l^^ de2iiP

Spherical harmonics 173

Higher dimensions

The classical spherical harmonics are defined as functions on the unit sphere S inside three-dimensional Euclidean
space. Spherical harmonics can be generalized to higher dimensional Euclidean space R" as follows. Let P

denote the space of homogeneous polynomials of degree ( in n variables. That is, a polynomial P is in P provided
that

F(Ax) = A'P(x).

obtained by restriction from A .

Let A denote the subspace of P consisting of all harmonic polynomials; these are the solid spherical harmonics. Let
H denote the space of functions on the unit sphere

S"- 1 = {xeR n | |x| = 1}

riction from A
The following properties hold:

n— 1

• The spaces H are dense in the set of continuous functions on S with respect to the uniform topology, by the

2 n-\

Stone-Weierstrass theorem. As a result, they are also dense in the space L (S ) of square-integrable functions on
the sphere.

• For all f G H „, one has

Asn-if = -e{£ + n-2)f.

n—l

where A n-1 is the Laplace-Beltrami operator on S . This operator is the analog of the angular part of the
Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as

V 2 = r l-.|_ r .-l|_ +r -2 As „_ i

or or

• It follows from the Stokes theorem and the preceding property that the spaces H are orthogonal with respect to

2 n-1
the inner product from L (S ). That is to say,

Js*

fgd^l =

for/G K ( and g € H^ for k * I.

• Conversely, the spaces H are precisely the eigenspaces of A n-1. In particular, an application of the spectral
theorem to the Riesz potential A—Tigives another proof that the spaces H are pairwise orthogonal and

2 n—l

complete in L (S ).

• Every homogeneous polynomial PGP can be uniquely written in the form

P{x) = P t {x) + |x| 2 F,_ 2 + ,

x\ £ P$ I even

xf-^P^x) I odd

where P G A . In particular,
; j

n— L J \ n — 1 /

An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of
separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian

9_ 9 _0 9 n

Ajn-i = sin n cj>— sin 71 0— + sin (j)A s »-2
d(j) dq>

n—l

where q) is the axial coordinate in a spherical coordinate system on S

Spherical harmonics 174

Connection with representation theory

The space H of spherical harmonics of degree ( is a representation of the symmetry group of rotations around a
point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on H
by function composition

ip i— > tp o p

for %p a spherical harmonic and p a rotation. The representation H is an irreducible representation of SO(3).

The elements of H arise as the restrictions to the sphere of elements of A : harmonic polynomials homogeneous of

degree I on three-dimensional Euclidean space R . By polarization of a|> G A , there are coefficients tp^ ^
symmetric on the indices, uniquely determined by the requirement

The condition that a|> be harmonic is equivalent to the assertion that the tensor ^ j must be trace free on every
pair of indices. Thus as an irreducible representation of SO(3), H is isomorphic to the space of traceless symmetric
tensors of degree (.

More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on the
n-sphere is the irreducible representation of SO(«+l) corresponding to the traceless symmetric ^-tensors. However,
whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in
higher dimensions have additional irreducible representations that do not arise in this manner.

The special orthogonal groups have additional spin representations that are not tensor representations, and are
typically not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are
representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions,
and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin
representations of SO(3), with respect to the action by quaternionic multiplication.

Generalizations

The angle-preserving symmetries of the two-sphere are described by the group of Mobius transformations PSL(2,C).
With respect to this group, the sphere is equivalent to the usual Riemann sphere. The group PSL(2,C) is isomorphic
to the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on the
celestial sphere in Minkowski space. The analog of the spherical harmonics for the Lorentz group is given by the
hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric
series, as SO(3) = PSU(2) is a subgroup of PSL(2,C).

More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in
particular, hypergeometric series can be developed for any Lie group.

Notes

[1] A historical account of various approaches to spherical harmonics in three-dimensions can be found in Chapter IV of MacRobert 1967. The
term "Laplace spherical harmonics" is in common use; see Courant & Hilbert 1962 and Meijer & Bauer 2004.

[2] The approach to spherical harmonics taken here is found in (Courant & Hilbert 1966, §V.8, §VII.5).

[3] Physical applications often take the solution that vanishes at infinity, making A = 0. This does not affect the angular portion of the spherical
harmonics.

[4] Edmonds 1957, §2.5

[5] Watson & Whittaker 1927, p. 392.

[6] This is valid for any orthonormal basis of spherical harmonics of degree I. For unit power harmonics it is necessary to remove the factor of

A-K-

[7] Watson & Whittaker 1927, p. 395

[8] Unsold 1927

[9] Stein & Weiss 1971, §IV.2

Spherical harmonics 175

[10] Eremenko, Jakobson & Nadirashvili 2007

[11] Solomentsev 2001; Stein & Weiss 1971, §Iv.2

[12] N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl., vol. 22, (1968).

[13] J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).

[14] W. Miller, Symmetry and Separation of Variables, Addison- Wesley, Reading (1977).

[15] A. Wawrzyriczyk, Group Representations and Special Functions, Polish Scientific Publishers. Warszawa (1984).

References

Cited references

Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume I, Wiley-Interscience.

Edmonds, A.R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press,

ISBN 0-691-07912-9.

Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai (2007), "On nodal sets and nodal domains on g 2

and ]g>2", Universitede Grenoble. Annales de llnstitut Fourier 57 (7): 2345-2360, ISSN 0373-0956,

MR2394544

MacRobert, T.M. (1967), Spherical harmonics: An elementary treatise on harmonic functions, with applications,

Pergamon Press.

Meijer, Paul Herman Ernst; Bauer, Edmond (2004), Group theory: The application to quantum mechanics,

Dover, ISBN 9780486437989.

Solomentsev, E.D. (2001), "Spherical harmonics" (http://eom.springer.de/S/s086690.htm), in Hazewinkel,

Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104.

Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.:

Princeton University Press, ISBN 978-0-691-08078-9.

Unsold, Albrecht (1927), "Beitrage zur Quantenmechanik der Atome", Annalen der Physik 387 (3): 355—393,

doi: 10. 1002/andp. 19273870304.

Watson, G. N; Whittaker, E. T. (1927), A Course of Modern Analysis, Cambridge University Press.

General references

E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, (1955) Chelsea Pub. Co., ISBN
978-0828401043.

C. Muller, Spherical Harmonics, (1966) Springer, Lecture Notes in Mathematics, Vol. 17, ISBN
978-3-540-03600-5.

E. U. Condon and G H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN
0-521-09209-4, See chapter 3.

J.D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X
Albert Messiah, Quantum Mechanics, volume II. (2000) Dover. ISBN 0-486-40924-4.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of 'Angular Momentum, (1988) World
Scientific Publishing Co., Singapore, ISBN 9971-5-0107-4

Weisstein, Eric W., " Spherical harmonics (http://mathworld.wolfram.com/SphericalHarmonic.html)" from
MathWorld.

Spherical harmonics 176

External links

• Interactive calculator of spherical harmonics on Tal Carmon's Research Homepage (http://wm.eecs. umich.
edu:8180/webMattiematica/tcarmon/sti2.jsp)

• Spherical harmonics applied to Acoustic Field analysis on Trinnov Audio's research page (http://www.trinnov.
com/en/ about-us/research/overview)

• Spherical Harmonics (http://demonstrations.wolfram.com/SphericalHarmonics/) by Stephen Wolfram and
Nodal Domains of Spherical Harmonics (http://demonstrations.wolfram.com/
NodalDomainsOfSphericalHarmonics/) by Michael Trott, the Wolfram Demonstrations Project

• An accessible introduction to spherical harmonics (by J. B. Calvert) (http://mysite.du.edu/~jcalvert/math/
harmonic/harmonic, htm)

• Citizendium:Spherical harmonics

• OpenGL Spherical harmonics demo (http://www.paulsprojects.net/opengl/sh/sh.html)

• Allen McNamara's spherical harmonics animations (http://mcnamara.asu.edu/sphere_harmonics/index.html)

• Thorsten Becker's spherical harmonics animations (http://geodynamics.usc.edu/~becker/teaching-sh.html)

Software

• Spherical harmonics generator in OpenGL (http://adomas.org/shg/)

• SHTOOLS: Fortran 95 software archive (http://www.ipgp.jussieu.fr/~wieczor/SHTOOLS/SHTOOLS.html)

• HEALPIX: Fortran 90 and C++ software archive (http://healpix.jpl.nasa.gov/)

• SpherePack: Fortran 77 software archive (http://www.cisl.ucar.edu/css/software/spherepack/)

• SpharmonicKit: C software archive (http://www.cs.dartmouth.edu/~geelong/sphere/)

• Frederik J Simons: Matlab software archive (http://geoweb.princeton.edu/people/simons/software.html)

• NFFT: C subroutine library (fast spherical Fourier transform for arbitrary nodes) (http://www-user.tu-chemnitz.
de/~potts/nfft/)

• Shansyn: spherical harmonics package for GMT/netcdf grd files (http://www.spice-rtn.org/library/software/
shansyn)

• SHAPE: Spherical HArmonic Parameterization Explorer (http://www.embl-heidelberg.de/~khairy/links.html)

Quantum computer

177

Quantum computer

A quantum computer is a device for computation that makes direct
use of quantum mechanical phenomena, such as superposition and
entanglement, to perform operations on data. Quantum computers are
different from traditional computers based on transistors. The basic
principle behind quantum computation is that quantum properties can
be used to represent data and perform operations on these data. A
theoretical model is the quantum Turing machine, also known as the
universal quantum computer.

Although quantum computing is still in its infancy, experiments have
been carried out in which quantum computational operations were
executed on a very small number of qubits (quantum bits). Both
practical and theoretical research continues, and many national
government and military funding agencies support quantum computing
research to develop quantum computers for both civilian and national

security purposes, such as cryptanalysis

[2]

The Bloch sphere is a representation of a qubit,

the fundamental building block of quantum

computers.

If large-scale quantum computers can be built, they will be able to solve certain problems much faster than any
classical computer using the best currently known algorithms (for example integer factorization using Shor's
algorithm or the simulation of quantum many-body systems). Furthermore, there exist quantum algorithms, such as
Simon's algorithm, which run exponentially faster than any possible probabilistic classical algorithm. Given
enough resources, a classical computer can simulate an arbitrary quantum computer. Hence, ignoring computational
and space constraints, a quantum computer is not capable of solving any problem which a classical computer
cannot.

Basis

A classical computer has a memory made up of bits, where each bit represents either a one or a zero. A quantum
computer maintains a sequence of qubits. A single qubit can represent a one, a zero, or, crucially, any quantum
superposition of these; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in
any superposition of 8. In general a quantum computer with n qubits can be in an arbitrary superposition of up to
2™ different states simultaneously (this compares to a normal computer that can only be in one of these 2 n states at
any one time). A quantum computer operates by manipulating those qubits with a fixed sequence of quantum logic
gates. The sequence of gates to be applied is called a quantum algorithm.

An example of an implementation of qubits for a quantum computer could start with the use of particles with two
spin states: "down" and "up" (typically written M \ and ||\ , or |fj\ and M\ ). But in fact any system possessing
an observable quantity A which is conserved under time evolution and such that A has at least two discrete and
sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. This is true because
any such system can be mapped onto an effective spin- 1/2 system.

Quantum computer

178

Bits vs. qubits

A quantum computer with a given number of qubits is exponentially more complex than a classical computer with
the same number of bits because describing the state of n qubits requires 2" complex coefficients. Measuring the
qubits would produce a classical state of only n bits, but such an action would also destroy the quantum state. We
can think of the system as being exactly one of the w-bit strings — we just do not know which one. For example, a

300 90

300-qubit quantum computer has a state described by 2 (approximately 10 ) complex numbers, more than the
number of atoms in the observable universe.

For example: Consider first a classical computer that operates on a
three-bit register. The state of the computer at any time is a
probability distribution over the 2 3 = gdifferent three-bit strings
000, 001, 010, 011, 100, 101, 110, 111. If it is a
deterministic computer, then it is in exactly one of these states
with probability 1 . However, if it is a probabilistic computer, then
there is a possibility of it being in any one of a number of different
states. We can describe this probabilistic state by eight
nonnegative numbers a,b,c,d,ef,g,h (where a = probability
computer is in state 0, b = probability computer is in state 01,
etc.). There is a restriction that these probabilities sum to 1.

6 «ii>

~|0)

|0101) «» |5)

|4> + |5)

qubits can be in a superposition of all the
clasically allowed states

Qubits are made up of controlled particles and the
means of control (e.g. devices that trap particles and
switch them from one state to another).

The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector (a,b,c,d,ef,g,h),
called a ket. However, instead of adding to one, the sum of the squares of the coefficient magnitudes,
| a | 2 _|_ Ifrl 2 _|_ ___ _|_ |/i| 2 , must equal one. Moreover, the coefficients are complex numbers. Since states are

represented by complex wavefunctions, two states being added together will undergo interference, which is a key
difference between quantum computing and probabilistic classical computing.

If you measure the three qubits, you will observe a three-bit string. The probability of measuring a given string is the
squared magnitude of that string's coefficient (i.e., the probability of measuring 00 = |a| 2 , the probability of

measuring 001 = Ifol 2 , etc.). Thus, measuring a quantum state described by complex coefficients (a,b,...,h) gives

the classical probability distribution (lal 2 |6| 2 ... |/i| 2 )and we say that the quantum state "collapses" to a

classical state as a result of making the measurement.

Note that an eight-dimensional vector can be specified in many different ways depending on what basis is chosen for

the space. The basis of bit strings (e.g., 000, 001, ..., Ill) is known as the computational basis. Other possible bases

are unit-length, orthogonal vectors and the eigenvectors of the Pauli-x operator. Ket notation is often used to make

the choice of basis explicit. For example, the state (a,b,c,d,ej,g,h) in the computational basis can be written as:

a |000) + b |001) + c |010) + d |011) + e 1 100) + / |101) + g |110) + h |111)
where, e.g., |010> = (0, 0, 1, 0, 0, 0, 0, 0)

The computational basis for a single qubit (two dimensions) is IfJ) = (1, 0)and |1) = (0, 1).

Using the eigenvectors of the Pauli-x operator, a single qubit is |+) = —m (1, ljand |— ) = -7? (1, — 1).

Quantum computer 179

Operation

While a classical three-bit state and a quantum three-qubit state are both eight-dimensional vectors, they are
manipulated quite differently for classical or quantum computation. For computing in either case, the system must be
initialized, for example into the all-zeros string, 1 000) , corresponding to the vector (1,0,0,0,0,0,0,0). In classical
randomized computation, the system evolves according to the application of stochastic matrices, which preserve that
the probabilities add up to one (i.e., preserve the LI norm). In quantum computation, on the other hand, allowed
operations are unitary matrices, which are effectively rotations (they preserve that the sum of the squares add up to
one, the Euclidean or L2 norm). (Exactly what unitaries can be applied depend on the physics of the quantum
device.) Consequently, since rotations can be undone by rotating backward, quantum computations are reversible.
(Technically, quantum operations can be probabilistic combinations of unitaries, so quantum computation really does
generalize classical computation. See quantum circuit for a more precise formulation.)

Finally, upon termination of the algorithm, the result needs to be read off. In the case of a classical computer, we
sample from the probability distribution on the three-bit register to obtain one definite three-bit string, say 000.
Quantum mechanically, we measure the three-qubit state, which is equivalent to collapsing the quantum state down
to a classical distribution (with the coefficients in the classical state being the squared magnitudes of the coefficients
for the quantum state, as described above) followed by sampling from that distribution. Note that this destroys the
original quantum state. Many algorithms will only give the correct answer with a certain probability, however by
repeatedly initializing, running and measuring the quantum computer, the probability of getting the correct answer
can be increased.

For more details on the sequences of operations used for various quantum algorithms, see universal quantum
computer, Shor's algorithm, Graver's algorithm, Deutsch-Jozsa algorithm, amplitude amplification, quantum Fourier
transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

Potential

Integer factorization is believed to be computationally infeasible with an ordinary computer for large integers if they
are the product of few prime numbers (e.g., products of two 300-digit primes). By comparison, a quantum
computer could efficiently solve this problem using Shor's algorithm to find its factors. This ability would allow a
quantum computer to decrypt many of the cryptographic systems in use today, in the sense that there would be a
polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the
popular public key ciphers are based on the difficulty of factoring integers (or the related discrete logarithm problem
which can also be solved by Shor's algorithm), including forms of RSA. These are used to protect secure Web pages,
encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic
privacy and security.

However, other existing cryptographic algorithms do not appear to be broken by these algorithms. Some

public -key algorithms are based on problems other than the integer factorization and discrete logarithm problems to
which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice

based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time
algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems,
is a well-studied open problem. It has been proven that applying Graver's algorithm to break a symmetric (secret
key) algorithm by brute force requires roughly 2 invocations of the underlying cryptographic algorithm, compared
with roughly 2 n in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would
have the same security against an attack using Graver's algorithm that AES-128 has against classical brute-force
search (see Key size). Quantum cryptography could potentially fulfill some of the functions of public key
cryptography.

Quantum computer 180

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the

ri3i

best known classical algorithm have been found for several problems, including the simulation of quantum
physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving
Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be
discovered, although this is considered unlikely. For some problems, quantum computers offer a polynomial
speedup. The most well-known example of this is quantum database search, which can be solved by Grover's
algorithm using quadratically fewer queries to the database than are required by classical algorithms. In this case the
advantage is provable. Several other examples of provable quantum speedups for query problems have subsequently
been discovered, such as for finding collisions in two-to-one functions and evaluating NAND trees.

Consider a problem that has these four properties:

1 . The only way to solve it is to guess answers repeatedly and check them,

2. The number of possible answers to check is the same as the number of inputs,

3. Every possible answer takes the same amount of time to check, and

4. There are no clues about which answers might be better: generating possibilities randomly is just as good as
checking them in some special order.

An example of this is a password cracker that attempts to guess the password for an encrypted file (assuming that the
password has a maximum possible length).

For problems with all four properties, the time for a quantum computer to solve this will be proportional to the
square root of the number of inputs. That can be a very large speedup, reducing some problems from years to
seconds. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret
key.

Grover's algorithm can also be used to obtain a quadratic speed-up over a brute-force search for a class of problems
known as NP-complete.

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to
simulate in an efficient manner classically, many believe quantum simulation will be one of the most important
applications of quantum computing.

There are a number of practical difficulties in building a quantum computer, and thus far quantum computers have
only solved trivial problems. David DiVincenzo, of IBM, listed the following requirements for a practical quantum

t [15]

computer:

• scalable physically to increase the number of qubits;

• qubits can be initialized to arbitrary values;

• quantum gates faster than decoherence time;

• universal gate set;

• qubits can be read easily.

Quantum decoherence

One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the
system from its environment as the slightest interaction with the external world would cause the system to decohere.
This effect is irreversible, as it is non-unitary, and is usually something that should be highly controlled, if not
avoided. Decoherence times for candidate systems, in particular the transverse relaxation time T (for NMR and MRI
technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an
often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio
of operating time to decoherence time, hence any operation must be completed much more quickly than the
decoherence time.

Quantum computer 181

If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors
due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time. An often

cited figure for required error rate in each gate is 10 . This implies that each gate must be able to perform its task in
one 10,000th of the decoherence time of the system.

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings
with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's

algorithm is still polynomial, and thought to be between L and L , where L is the number of bits in the number to be
factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number,
this implies a need for about 10 qubits without error correction. With error correction, the figure would rise to

■7

about 10 qubits. Note that computation time is about £ 2 or about ^Q 7 steps and on 1 MHz, about 10 seconds.

A very different approach to the stability-decoherence problem is to create a topological quantum computer with

n 7i n si
anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

Developments

There are a number of quantum computing models, distinguished by the basic elements in which the computation is
decomposed. The four main models of practical importance are

• the quantum gate array (computation decomposed into sequence of few-qubit quantum gates),

• the one-way quantum computer (computation decomposed into sequence of one-qubit measurements applied to a
highly entangled initial state (cluster state)),

• the adiabatic quantum computer (computation decomposed into a slow continuous transformation of an initial
Hamiltonian into a final Hamiltonian, whose ground states contains the solution),

• and the topological quantum computer (computation decomposed into the braiding of anyons in a 2D lattice)

The Quantum Turing machine is theoretically important but direct implementation of this model is not pursued. All
four models of computation have been shown to be equivalent to each other in the sense that each can simulate the
other with no more than polynomial overhead.

For physically implementing a quantum computer, many different candidates are being pursued, among them
(distinguished by the physical system used to realize the qubits):

• Superconductor-based quantum computers (including SQUID-based quantum computers) (qubit
implemented by the state of small superconducting circuits (Josephson junctions))

• Trapped ion quantum computer (qubit implemented by the internal state of trapped ions)

• Optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)

T221

• electrically-defined or self-assembled quantum dots (e.g. the Loss-DiVincenzo quantum computer or ) (qubit

given by the spin states of an electron trapped in the quantum dot)

• Quantum dot charge based semiconductor quantum computer (qubit is the position of an electron inside a double
quantum dot)

• Nuclear magnetic resonance on molecules in solution (liquid-state NMR) (qubit provided by nuclear spins within
the dissolved molecule)

• Solid-state NMR Kane quantum computers (qubit realized by the nuclear spin state of phosphorus donors in
silicon)

• Electrons-on-helium quantum computers (qubit is the electron spin)

• Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of atoms trapped in and coupled to
high-finesse cavities)

• Molecular magnet

• Fullerene-based ESR quantum computer (qubit based on the electronic spin of atoms or molecules encased in
fullerene structures)

Quantum computer

182

• Optics-based quantum computer (Quantum optics) (qubits realized by appropriate states of different modes of the

T241
electromagnetic field, e.g. )

T9S1 T9A1 f771

• Diamond-based quantum computer (qubit realized by the electronic or nuclear spin of
Nitrogen-vacancy centers in diamond)

T281

• Bose— Einstein condensate-based quantum computer

• Transistor-based quantum computer - string quantum computers with entrainment of positive holes using an
electrostatic trap

[291 [301

• Rare-earth-metal-ion-doped inorganic crystal based quantum computers (qubit realized by the internal
electronic state of dopants in optical fibers)

The large number of candidates demonstrates that the topic, in spite of rapid progress, is still in its infancy. But at the
same time, there is also a vast amount of flexibility.

In 2005, researchers at the University of Michigan built a semiconductor chip which functioned as an ion trap. Such
devices, produced by standard lithography techniques, may point the way to scalable quantum computing tools.
An improved version was made in 2006.

In 2009, researchers at Yale University created the first rudimentary solid-state quantum processor. The two-qubit
superconducting chip was able to run elementary algorithms. Each of the two artificial atoms (or qubits) were made

[32] [331

up of a billion aluminum atoms but they acted like a single one that could occupy two different energy states.

Another team, working at the University of Bristol, also created a silicon-based quantum computing chip, based on

T341
quantum optics. The team was able to run Shor's algorithm on the chip. The latest developments [for 2010] can be

T351
found in. Springer publish a Journal devoted to the subject [36].

A team of scientists from Australia and Japan have finally made a breakthrough in quantum teleportation. They have
successfully transferred a complex set of quantum data with full transmission integrity achieved. Also the qubits

being destroyed in one place but instantaneously resurrected in another, without affecting their superpositions

[37]

Relation to computational complexity theory

The class of problems that can be efficiently solved by quantum

computers is called BQP, for "bounded error, quantum, polynomial

time". Quantum computers only run probabilistic algorithms, so BQP

on quantum computers is the counterpart of BPP ("bounded error,

probabilistic, polynomial time") on classical computers. It is defined as

the set of problems solvable with a polynomial-time algorithm, whose

[39]
probability of error is bounded away from one half. A quantum

computer is said to "solve" a problem if, for every instance, its answer

will be right with high probability. If that solution runs in polynomial

time, then that problem is in BQP.

PSPACE problems

The suspected relationship of BQP to other
problem spaces.

BQP is contained in the complexity class #P (or more precisely in the
associated class of decision problems P ), which is a subclass of
PSPACE.

BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. Both integer
factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and
hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum
computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally
suspected to be false.

Possibilities of the quantum computer to accelerate classical algorithms has rigid limits — upper bounds of quantum
computation's complexity. The overwhelming part of classical calculations cannot be accelerated on the quantum

Quantum computer 183

[411
computer. A similar fact takes place for particular computational tasks, like the search problem, for which

[421
Grover's algorithm is optimal.

Although quantum computers may be faster than classical computers, those described above can't solve any
problems that classical computers can't solve, given enough time and memory (however, those amounts might be
practically infeasible). A Turing machine can simulate these quantum computers, so such a quantum computer could

never solve an undecidable problem like the halting problem. The existence of "standard" quantum computers does

T431
not disprove the Church— Turing thesis. It has been speculated that theories of quantum gravity, such as M-theory

or loop quantum gravity, may allow even faster computers to be built. Currently, it's an open problem to even define

computation in such theories due to the problem of time, i.e. there's no obvious way to describe what it means for an

[441
observer to submit input to a computer and later receive output.

References

[I] " Quantum Computing with Molecules (http://phm.cba.mit.edu/papers/98.06.sciam/0698gershenfeld.html)" article in Scientific
American by Neil Gershenfeld and Isaac L. Chuang

[2] Quantum Information Science and Technology Roadmap (http://qist.lanl.gov/qcomp_map.shtml) for a sense of where the research is

heading.
[3] Simon, D.R. (1994), "On the power of quantum computation", Foundations of Computer Science, 1994 Proceedings., 35th Annual

Symposium on: 116—123
[4] Nielsen, Michael A.; Chuang, Isaac L.. Quantum Computation and Quantum Information, p. 202.
[5] Waldner, Jean-Baptiste (2007). Nanocomputers and Swarm Intelligence. London: ISTE. p. 157. ISBN 2746215160.
[6] David P. DiVincenzo (1995). "Quantum Computation". Science 270 (5234): 255-261. Bibcode 1995Sci...270..255D.

doi:10.1126/science.270.5234.255.
[7] Integer Factoring (http://modular.fas.harvard.edu/edu/Fall2001/124/misc/arjen_lenstra_factoring.pdf) By ARJEN K. LENSTRA -

Designs, Codes and Cryptography, 19, 101—128 (2000) Kluwer Academic Publishers
[8] Daniel J. Bernstein, Introduction to Post-Quantum Cryptography (http://pqcrypto.org/www.springer.com/cda/content/document/

cda_downloaddocument/9783540887010-cl.pdf). Introduction to Daniel J. Bernstein, Johannes Buchmann, Erik Dahmen (editors).

Post-quantum cryptography. Springer, Berlin, 2009. ISBN 978-3-540-88701-0
[9] See also pqcrypto.org (http://pqcrypto.org/), a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known

to be broken by quantum computing.
[10] Robert J. McEliece. " A public-key cryptosystem based on algebraic coding theory (http://ipnpr.jpl.nasa.gov/progress_report2/42-44/

44N.PDF)." Jet Propulsion Laboratory DSN Progress Report 42-44, 1 14-1 16.

[II] Kobayashi, H.; Gall, F.L. (2006). "Dihedral Hidden Subgroup Problem: A Survey" (http://www.jstage.jst.go.Jp/article/imt/l/l/l_178/
_article). Information and Media Technologies 1 (1): 178—185. .

[12] Bennett C.H., Bernstein E., Brassard G., Vazirani U., The strengths and weaknesses of quantum computation (http://www.es. berkeley.

edu/~vazirani/pubs/bbbv.ps). SIAM Journal on Computing 26(5): 1510-1523 (1997).
[13] Quantum Algorithm Zoo (http://www.its.caltech.edu/~sjordan/zoo.html) - Stephen Jordan's Homepage
[14] The Father of Quantum Computing (http://www.wired.com/science/discoveries/news/2007/02/72734) By Quinn Norton 02.15.2007,

Wired.com
[15] David P. DiVincenzo, IBM (2000-04-13). "The Physical Implementation of Quantum Computation". arXiv:quant-ph/0002077.
[16] M. I. Dyakonov, Universite Montpellier (2006-10-14). "Is Fault-Tolerant Quantum Computation Really Possible?".

arXiv:quant-ph/06101 17.
[17] Freedman, Michael; Alexei Kitaev, Michael Larsen, Zhenghan Wang (2002-10-20). "Topological Quantum Computation". Bulletin of the

American Mathematical Society 40 (1): 31-38. doi:10.1090/S0273-0979-02-00964-3.
[18] Monroe, Don, Anyons: The breakthrough quantum computing needs? (http://www.newscientist.com/channel/fundamentals/

mg20026761.700-anyons-the-breakthrough-quantum-computing-needs.html), New Scientist, 1 October 2008
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an 'input' and 'later' receive an 'output' — there is no such thing as computation, not even theoretically." (emphasis in original)

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Quantum computer 185

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David P. DiVincenzo (1995). "Quantum Computation". Science 270 (5234): 255-261.

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Gregg Jaeger (2006). Quantum Information: An Overview. Berlin: Springer. ISBN 0-387-35725-4.

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Stephanie Frank Singer (2005). Linearity, Symmetry, and Prediction in the Hydrogen Atom. New York: Springer.

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for Quantum Computation. arXiv:quant-ph/0002077.

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csee. umbc. edu/~lomonaco/Lectures .html#OxfordLectures)

C. Adami, N.J. Cerf. (1998). "Quantum computation with linear optics". arXiv:quant-ph/9806048vl.

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Rolf Landauer, (1961). "Irreversibility and heat generation in the computing process" (http://www. research.
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Gordon E. Moore (1965). Cramming more components onto integrated circuits.

R.W. Keyes, (1988). Miniaturization of electronics and its limits.

M. A. Nielsen,; E. Knill, ; R. Laflamme,. "Complete Quantum Teleportation By Nuclear Magnetic Resonance"
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Lieven M.K. Vandersypen,; Constantino S. Yannoni, ; Isaac L. Chuang, (2000). Liquid state NMR Quantum
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Imai Hiroshi,; Hayashi Masahito, (2006). Quantum Computation and Information. Berlin: Springer.
ISBN 3540331328.

Andre Berthiaume, (1997). "Quantum Computation" (http://citeseer.ist.psu.edu/article/berthiaume97quantum.
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Daniel R. Simon, (1994). "On the Power of Quantum Computation" (http://citeseer.ist.psu.edu/simon94power.
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communication security at the Ruhr-University Bochum.

Laura Sanders, (2009). "First programmable quantum computer created" (http://www.sciencenews.org/view/

generic/id/49951/title/First_programmable_quantum_computer_created).

"NEW TRENDS IN QUANTUM COMPUTATION" (http://insti.physics.sunysb.edu/itp/conf/

simons-qcomputation2/).

Quantum computer 186

External links

• Stanford Encyclopedia of Philosophy: " Quantum Computing (http://plato.stanford.edu/entries/qt-quantcomp/
)" by Amit Hagar.

• Quantiki (http://www.quantiki.org/) - Wiki and portal with free-content related to quantum information
science.

• jQuantum: Java quantum circuit simulator (http://jquantum.sourceforge.net/jQuantumApplet.html)

• QCAD: Quantum circuit emulator (http://www.phys.cs.is.nagoya-u.ac.jp/~watanabe/qcad/index.html)

• C++ Quantum Library (http://gna.org/projects/quantumlibrary)

• QLISP Project: Quantum Programming Language (http://www.schloerconsulting.com/
quantum-computer-q-lisp-programming-language)

• Haskell Library for Quantum computations (http://hackage.haskell.org/cgi-bin/hackage-scripts/package/
quantum-arrow)

• Video Lectures by David Deutsch (http://www.quiprocone.org/Protected/DD_lectures.htm)

• Lectures at the Institut Henri Poincare (slides and videos) (http://www.quantware.ups-tlse.fr/IHP2006/)

• Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008) (http://nanohub.org/
resources/4778)

• Online Web-based Quantum Computer Simulator (University Of Patras, Wire Communications Laboratory)
(http://www.wcl.ece.upatras.gr/ai/resources/demo-quantum-simulation)

Topological Quantum Computers

A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles
called anyons, whose world lines cross over one another to form braids in a three-dimensional spacetime (i.e., one
temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage
of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much
more stable. The smallest perturbations can cause a quantum particle to decohere and introduce errors in the
computation, but such small perturbations do not change the topological properties of the braids. This is like the
effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an
ordinary quantum particle in four-dimensional spacetime) simply bumping into a wall. While the elements of a
topological quantum computer originate in a purely mathematical realm, recent experiments indicate these elements
can be created in the real world using semiconductors made of gallium arsenide near absolute zero and subjected to
strong magnetic fields.

Introduction

Anyons are quasiparticles in a two-dimensional space. Anyons are not strictly fermions or bosons, but do share the
characteristic of fermions in that they cannot occupy the same state. Thus, the world lines of two anyons cannot cross
or merge. This allows braids to be made that make up a particular circuit. In the real world, anyons form from the
excitations in an electron gas in a very strong magnetic field, and carry fractional units of magnetic flux in a
particle-like manner. This phenomenon is called the fractional quantum Hall effect. The electron "gas" is sandwiched
between two flat plates of gallium arsenide, which create the two-dimensional space required for anyons, and is
cooled and subjected to intense transverse magnetic fields.

When anyons are braided, the transformation of the quantum state of the system depends only on the topological
class of the anyons' trajectories (which are classified according to the braid group). Therefore, the quantum
information which is stored in the state of the system is impervious to small errors in the trajectories. In 2005, Sankar
Das Sarma, Michael Freedman, and Chetan Nayak proposed a quantum Hall device which would realize a

Topological Quantum Computers 187

topological qubit. In a key development for topological quantum computers, in 2005 Vladimir J. Goldman, Fernando
E. Camino, and Wei Zhou were said to have created the first experimental evidence for using fractional quantum
Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena
not involving anyons. It should also be noted that nonabelian anyons, a species required for topological quantum
computers, have yet to be experimentally confirmed.

The original proposal for topological quantum computation is due to Alexei Kitaev in 1997.

Topological vs. standard quantum computer

Topological quantum computers are equivalent in computational power to other standard models of quantum
computation, in particular to the quantum circuit model and to the quantum Turing machine model. That is, any of
these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to
the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first
developed in the topological model, and only later converted and extended in the standard quantum circuit model.

Computations

To live up to its name, a topological quantum computer must provide the unique computation properties promised by
a conventional quantum computer design, which uses trapped quantum particles. Fortunately in 2002, Michael H.
Freedman along with Zhenghan Wang, both with Microsoft, and Michael Larsen of Indiana University proved that a
topological quantum computer can, in principle, perform any computation that a conventional quantum computer can
do.

They found that conventional quantum computer device, given a flawless (error-free) operation of its logic circuits,
will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with
flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the
answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple
linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of
accuracy in the answer. Actual computation [gates] are done by edge states of fractional quantum Hall effect. This
make models one dimensional anyons important. In one space dimension anyons are defined algebraically.

Error correction and control

Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control
for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining
braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering
strays drops to near zero. It has been estimated that the error rate for a logical NOT operation of a qubit state could

—30

be as low as 10 or less. Although this number has been criticized as being strongly overstated, there is nonetheless
good reason to believe that topologically protected systems will be particularly immune to many sources of error that
plague other schemes for quantum information processing.

Simulating the dynamics of a topological quantum computer may be a promising method of implementing
fault-tolerant quantum computation even with a standard quantum information processing scheme. Raussendorf,
Harrington, and Goyal have studied one model, with promising simulation results.

Topological Quantum Computers 188

References

• "Computing with Quantum Knots" Graham P. Collins, Scientific American, April 2006. [1]

• "Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State", Sankar Das
Sarma, Michael Freedman, and Chetan Nayak [2], Phys. Rev. Lett. 94, 166802 (2005).

• "Non-Abelian Anyons and Topological Quantum Computation", Chetan Nayak [2], Steven H. Simon, Ady Stern,
Michael Freedman, Sankar Das Sarma, Rev. Mod. Phys. 80, 1083 (2008); http://www.arxiv.org/abs/0707.
1889, 2007

• "Topological fault-tolerance in cluster state quantum computation", Robert Raussendorf, Jim Harrington, Kovid
Goyal, http://arxiv.org/abs/quant-ph/0703143, 2007

References

[1] http://info.phys.unm.edu/~thedude/topo/sciamTQC.pdf
[2] http://stationq.cnsi.ucsb.edu/~nayak/

189

Categorical and Topological Dynamics.
Category Theory and Categorical Dynamics

Concepts

Algebraic Geometry

Algebraic geometry is a branch of mathematics which combines
techniques of abstract algebra, especially commutative algebra, with
the language and the problems of geometry. It occupies a central place
in modern mathematics and has multiple conceptual connections with
such diverse fields as complex analysis, topology and number theory.
Initially a study of systems of polynomial equations in several
variables, the subject of algebraic geometry starts where equation
solving leaves off, and it becomes even more important to understand
the intrinsic properties of the totality of solutions of a system of
equations, than to find some solution; this leads into some of the
deepest waters in the whole of mathematics, both conceptually and in
terms of technique.

This Togliatti surface is an algebraic surface of
degree five.

The fundamental objects of study in algebraic geometry are algebraic

varieties, geometric manifestations of solutions of systems of

polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals,

form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its

coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and

relations between the curves given by different equations.

Descartes's idea of coordinates is central to algebraic geometry, but it has undergone a series of remarkable
transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real
numbers, but this changed when first complex numbers, and then elements of an arbitrary field became acceptable.
Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in a
different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in
the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on
'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an
ambient coordinate space; this parallels developments in topology and complex geometry.

One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the
form given to it by Grothendieck and Serre, is that the former is concerned with the more geometric notion of a
point, while the latter emphasizes the more analytic concepts of a regular function and a regular map and extensively
draws on sheaf theory. Another important difference lies in the scope of the subject. Grothendieck' s idea of scheme
provides the language and the tools for geometric treatment of arbitrary commutative rings and, in particular, bridges
algebraic geometry with algebraic number theory. Andrew Wiles's celebrated proof of Fermat's last theorem is a
vivid testament to the power of this approach. Andre Weil, Grothendieck, and Deligne also demonstrated that the
fundamental ideas of topology of manifolds have deep analogues in algebraic geometry over finite fields.

Algebraic Geometry

190

Zeros of simultaneous polynomials

In classical algebraic geometry, the main objects of interest are the
vanishing sets of collections of polynomials, meaning the set of all
points that simultaneously satisfy one or more polynomial
equations. For instance, the two-dimensional sphere in

three-dimensional Euclidean space R could be defined as the set
of all points (x,y,z) with

* 2 + y 2 + z 2 = 1

x+y+z=0

Sphere and slanted circle

2 , 2,2

x +y + z

1 = 0.

A "slanted" circle in R can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations

x 2 + y 2 + z 2
x + y + z = 0.

1 = 0,

Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many
of the same results are true if we assume only that k is algebraically closed. We define A (k) (or more simply A",
when k is clear from the context), called the affine n-space over k, to be lc . The purpose of this apparently
superfluous notation is to emphasize that one 'forgets' the vector space structure that lc carries. Abstractly speaking,
A" is, for the moment, just a collection of points.

n 1

A function/ : A — > A is said to be regular if it can be written as a polynomial, that is, if there is a polynomial p in
k[x ,...,x ] such that/(f ,...,? )=p{t .,...,? ) for every point (t ,...,t ) of A".

Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will refer to
the set of all regular functions on A" as k[A\.

We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in
k[A ]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in A' where every polynomial in S
vanishes. In other words,

V{S) = {fa, . . . , t n )\Vp G S,p{t 1: ...,t n ) = 0}.

A subset of A ' which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of
algebraic set to be defined below).

Given a subset U of A , can one recover the set of polynomials which generate it? If U is any subset of A , define
I(U) to be the set of all polynomials whose vanishing set contains U. The / stands for ideal: if two polynomials /and
g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanishes on U, so I(U) is always an
ideal of k[A n ].

Two natural questions to ask are:

• Given a subset U of A", when is U= V(I(U))1

Algebraic Geometry 191

• Given a set S of polynomials, when is S = I(V(S))1

The answer to the first question is provided by introducing the Zariski topology, a topology on A which directly
reflects the algebraic structure of k[A n ]. Then U = V(I(U)) if and only if U is a Zariski-closed set. The answer to the
second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I(V(S)) is the prime radical of
the ideal generated by S. In more abstract language, there is a Galois connection, giving rise to two closure operators;
they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection.

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U.
Hilbert's basis theorem implies that ideals in k[A n ] are always finitely generated.

An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible
algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials
defining it generate a prime ideal of the polynomial ring.

Regular functions

Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on
differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular
function on an algebraic set V contained in A n is defined to be the restriction of a regular function on A n , in the
sense we defined above.

It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is
very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a
continuous function on a closed subset always extends to the ambient topological space.

Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V\.
This ring is called the coordinate ring of V.

Since regular functions on V come from regular functions on A n , there should be a relationship between their
coordinate rings. Specifically, to get a function in k[V\ we took a function in k[A ], and we said that it was the same
as another function if they gave the same values when evaluated on V. This is the same as saying that their difference
is zero on V. From this we can see that k[V\ is the quotient k[A ]/I(V).

The category of affine varieties

Using regular functions from an affine variety to A , we can define regular functions from one affine variety to
another. First we will define a regular function from a variety into affine space: Let V be a variety contained in A .
Choose m regular functions on V, and call them/ , ...,/ . We define a regular function / from V to A by letting

f(f,, ..., t ) = (f,, ..., f ). In other words, each f determines one coordinate of the range of f.

J 1 n v l J m J i o j

If V is a variety contained in A , we say that/is a regular function from V to V if the range of/is contained in V.

This makes the collection of all affine varieties into a category, where the objects are affine varieties and the
morphisms are regular maps. The following theorem characterizes the category of affine varieties:

The category of affine varieties is the opposite category to the category of finitely generated integral fc-algebras
and their homomorphisms.

Algebraic Geometry

192

Projective space

Consider the variety V(y - x ). If we draw it,
we get a parabola. As x increases, the slope

of the line from the origin to the point (x, x )
becomes larger and larger. As x decreases,
the slope of the same line becomes smaller
and smaller.

Compare this to the variety V(y - x ). This
is a cubic equation. As x increases, the slope

of the line from the origin to the point (x, x )
becomes larger and larger just as before. But
unlike before, as x decreases, the slope of
the same line again becomes larger and
larger. So the behavior "at infinity" of

V(y - x') is different from the behavior "at

infinity" of V(y - x ). It is, however,
difficult to make the concept of "at infinity"
meaningful, if we restrict to working in affine space.

The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact
Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The

behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y - x ) has a

singularity at one of those extra points, but V(y - x ) is smooth.

While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates
allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many
theorems in algebraic geometry simpler and sharper: For example, Bezout's theorem on the number of intersection
points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective
space plays a fundamental role in algebraic geometry.

2 3

parabola (y = x , red) and cubic (y = x , blue) in projective space

The modern viewpoint

The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various
levels of generality to schemes, formal schemes, ind-schemes, algebraic spaces, algebraic stacks and so on. The need
for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be
accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs,
constructing quotients by group actions and developing formal grounds for natural intersection theory and
deformation theory lead to some of the further extensions.

Most remarkably, in late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a
scheme. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a
category which is antiequivalent to the category of commutative unital rings, extending the duality between the
category of affine algebraic varieties over a field k, and the category of finitely generated reduced fc-algebras. The
gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the
Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The
Zariski topology in the set theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck
topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and
more sensitive examples than the crude Zariski topology, namely the etale topology, and the two flat Grothendieck
topologies: ffpf and fpqc; nowadays some other examples became prominent including Nisnevich topology. Sheaves

Algebraic Geometry 193

can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability
conditions leading to Artin stacks and, even finer, Deligne-Mumford stacks, both often called algebraic stacks.

Sometimes other algebraic sites replace the category of affine schemes. For example, Nikolai Durov has introduced
commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a
tropical geometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's
geometry were realized in this setup.

Another formal generalization is possible to Universal algebraic geometry in which every variety of algebra has its
own algebraic geometry. The term variety of algebra should not be confused with algebraic variety.

The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric
concepts and became cornerstones of modern algebraic geometry.

Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection
theory, this is often the most natural approach. One can extend the Grothendieck site of affine schemes to a higher
categorical site of derived affine schemes, by replacing the commutative rings with an infinity category of
differential graded commutative algebras, or of simplicial commutative rings or a similar category with an
appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial
sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of
derived stack as such a presheaf on the infinity category of derived affine schemes, which is satifsying certain
infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability
conditions). Quillen model categories, Segal categories and quasicategories are some of the most often used tools to
formalize this yielding the derived algebraic geometry, introduced by the school of Carlos Simpson, including
Andre Hirschowitz, Bertrand Toen, Gabrielle Vezzosi, Michel Vaquie and others; and developed further by Jacob
Lurie, Bertrand Toen, and Gabrielle Vezzosi. Another (noncommutative) version of derived algebraic geometry,
using A-infinity categories has been developed from early 1990-s by Maxim Kontsevich and followers.

History

Prehistory: Before the 19th century

Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC.
The Delian problem, for instance, was to construct a length x so that the cube of side x contained the same volume as

the rectangular box a b for given sides a and b. Menechmus (circa 350 BC) considered the problem geometrically by
intersecting the pair of plane conies ay = x and xy = ab. The later work, in the 3rd century BC, of Archimedes and
Apollonius studied more systematically problems on conic sections, and also involved the use of coordinates.
The Arab mathematicians were able to solve by purely algebraic means certain cubic equations, and then to interpret
the results geometrically. This was done, for instance, by Ibn al-Haytham in the 10th century AD. Subsequently,
Persian mathematician Omar Khayyam (born 1048 A.D.) discovered the general method of solving cubic equations
by intersecting a parabola with a circle. Each of these early developments in algebraic geometry dealt with
questions of finding and describing the intersections of algebraic curves.

Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of
Renaissance mathematicians such as Gerolamo Cardano and Niccolo Fontana "Tartaglia" on their studies of the
cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by
most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and
analytical methods in geometry. The French mathematicians Franciscus Vieta and later Rene Descartes and Pierre
de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of
coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by
Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on
conies and cubics (in the case of Descartes).

Algebraic Geometry 194

During the same period, Blaise Pascal and Gerard Desargues approached geometry from a different perspective,
developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the
purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimately, the analytic
geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete
quantitative tools needed to study physical problems using the new calculus of Newton and Leibniz. However, by the
end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the calculus of
infinitesimals of Lagrange and Euler.

Nineteenth and early 20th century

It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals in order to
bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by
Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties of projective
space. Cayley introduced the idea of homogeneous polynomial forms, and more specifically quadratic forms, on
projective space. Subsequently, Felix Klein studied projective geometry (along with other sorts of geometry) from
the viewpoint that the geometry on a space is encoded in a certain class of transformations on the space. By the end
of the 19th century, projective geometers were studying more general kinds of transformations on figures in
projective space. Rather than the projective linear transformations which were normally regarded as giving the
fundamental Kleinian geometry on projective space, they concerned themselves also with the higher degree
birational transformations. This weaker notion of congruence would later lead members of the 20th century Italian
school of algebraic geometry to classify algebraic surfaces up to birational isomorphism.

The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the
development of Riemann surfaces.

Twentieth century

B. L. van der Waerden, Oscar Zariski, Andre Weil and others attempted to develop a rigorous foundation for
algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals.

In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf
theory. Later, from about 1960, and largely spearheaded by Grothendieck, the idea of schemes was worked out, in
conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field
stabilized in the 1970s, and new applications were made, both to number theory and to more classical geometric
questions on algebraic varieties, singularities and moduli.

An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties,
which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic
curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in
elliptic curve cryptography.

While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for
effective computation with concretely-given polynomials have also been developed. The most important is the
technique of Grobner bases which is employed in all computer algebra systems. Based on these methods, several
solvers may compute all the solutions of a system of polynomial equations whose associated variety has dimension
zero and thus consists in a finite number of points.

Algebraic Geometry 195

Applications

Algebraic geometry now finds application in statistics, control theory, robotics, error-correcting codes,
phylogenetics and geometric modelling. There are also connections to string theory, game theory, graph
matchings, solitons and integer programming. Google Scholar lists hundreds of more studies on algebraic
geometry in biology , chemistry , economics , physics and of course other areas of mathematics .

Notes

[I] Dieudonne, Jean (1972). "The historical development of algebraic geometry" (http://jstor.org/stable/2317664). The American
Mathematical Monthly (The American Mathematical Monthly, Vol. 79, No. 8) 79 (8): 827-866. doi: 10.2307/23 17664.

[2] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, pp. 108, 90.
[3] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, p. 193.
[4] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, pp. 193—195.
[5] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, p. 279.
[6] Mathias Drton, Bernd Sturmfels, Seth Sullivant (2009), Lectures on Algebraic Statistics (http://books.google.co.uk/

books?id=TytYUTy5V_IC) Springer, ISBN 9783764389048
[7] Peter L. Falb (1990), Methods of algebraic geometry in control theory (http://books. google. co.uk/books?id=oJGYPwAACAAJ&),

Birkhauser, ISBN 9783764334543
[8] J. M. Selig (205), Geometric fundamentals of robotics (http://books. google. co.uk/books ?id=GUuzEOWilOQC), Springer, ISBN

9780387208749
[9] Michael A. Tsfasman, Serge G. VladuJ, Dmitry Nogin (2007), Algebraic geometric codes: basic notions (http://books.google.co.uk/

books?id=o2sA-wzDBLkC), AMS Bookstore, ISBN 9780821843062
[10] Barry A. Cipra (2007), Algebraic Geometers See Ideal Approach to Biology (http://siam.org/pdf/news/1146.pdf), SIAM News, Volume

40, Number 6

[II] Bert Juttler, Ragni Piene (2007) Geometric modeling and algebraic geometry (http://books. google. co.uk/books 7id=lwNGq87gWykC),
Springer, ISBN 9783540721840

[12] David A. Cox, Sheldon Katz (1999) Mirror symmetry and algebraic geometry (http://books. google. co.uk/books?id=vwL4ZewC81MC),

AMS Bookstore, ISBN 9780821821275
[13] The algebraic geometry of perfect and sequential equilibrium (http://econwpa.wustl.edu/econ-wp/game/papers/9309/9309001.pdf), LE

Blume, WRZame - Econometrica: Journal of the Econometric Society, 1994 -jstor.org
[14] Richard Kenyon, Andrei Okounkov, Scott Sheffield (2003) Dimers and Amoebae (http://arxiv.org/abs/math-ph/031 1005vl)
[15] IM Krichever and PG Grinevich, Algebraic geometry methods in soliton theory, Chapter 14 of Soliton theory (http://books.google.co.uk/

books?id=eO_PAAAAIAAJ), Allan P. Fordy, Manchester University Press ND, 1990, ISBN 9780719014918
[16] David A. Cox, Bernd Sturmfels, Dinesh N. Manocha (1997 Applications of computational algebraic geometry (http://books.google.co.uk/

books?id=feOMJEPDwzAC), AMS Bookstore, ISBN 9780821807507
[17] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=bio
[18] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=chm
[19] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=bus
[20] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=phy
[21] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=eng

References

A classical textbook, predating schemes:

• W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 1. Cambridge University Press.
ISBN 0-521-46900-7.

• W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 2. Cambridge University Press.
ISBN 0-521-46901-5.

• W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 3. Cambridge University Press.
ISBN 0-521-46775-6.

Modern textbooks that do not use the language of schemes:

• David A. Cox; John Little, Donal O'Shea (1997). Ideals, Varieties, and Algorithms (second ed.). Springer- Verlag.
ISBN 0-387-94680-2.

• Phillip Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.

Algebraic Geometry 196

• Joe Harris (1995). Algebraic Geometry: A First Course. Springer- Verlag. ISBN 0-387-97716-3.

• David Mumford (1995). Algebraic Geometry I: Complex Projective Varieties (2nd ed.). Springer- Verlag.
ISBN 3-540-58657-1.

• Miles Reid (1988). Undergraduate Algebraic Geometry. Cambridge University Press. ISBN 0-521-35662-8.

• Igor Shafarevich (1995). Basic Algebraic Geometry I: Varieties in Projective Space (2nd ed.). Springer- Verlag.
ISBN 0-387-54812-2.

Textbooks and references for schemes:

• David Eisenbud; Joe Harris (1998). The Geometry of Schemes. Springer- Verlag. ISBN 0-387-98637-5.

• Alexander Grothendieck (1960). Elements de geometrie algebrique. Publications mathematiques de 1'IHES.

• Alexander Grothendieck (1971). Elements de geometrie algebrique. 1 (2nd ed.). Springer- Verlag.
ISBN 3-540-05113-9.

• Robin Hartshorne (1977). Algebraic Geometry. Springer- Verlag. ISBN 0-387-90244-9.

• David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on
Curves and Their Jacobians (2nd ed.). Springer- Verlag. ISBN 3-540-63293-X.

• Igor Shafarevich (1995). Basic Algebraic Geometry II: Schemes and Complex Manifolds. Springer- Verlag.
ISBN 0-387-54812-2.

On the Internet:

• Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System (http://odin.mdacc.tmc.edu/~krc/
agathos/). In construction; currently of very limited use for self study.

• Algebraic geometry (http://planetmath.org/encyclopedia/AlgebraicGeometry.html) entry on PlanetMath (http:/
/planetmath.org/)

• Algebraic Equations and Systems of Algebraic Equations (http://eqworld.ipmnet.ru/en/solutions/ae.htm) at
Eq World: The World of Mathematical Equations

Category theory

197

Category theory

Category theory is an area of study in mathematics that examines
in an abstract way the properties of particular mathematical
concepts, by formalising them as collections of objects and arrows
(also called morphisms, although this term also has a specific, non
category-theoretical sense), where these collections satisfy certain
basic conditions. Many significant areas of mathematics can be
formalised as categories, and the use of category theory allows
many intricate and subtle mathematical results in these fields to be
stated, and proved, in a much simpler way than without the use of
categories.

The most accessible example of a category is the category of sets,

where the objects are sets and the arrows are functions from one

set to another. However it is important to note that the objects of a a category with objects X, Y, Z and morphisms/; g

category need not be sets nor the arrows functions; any way of

formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is

a valid category, and all the results of category theory will apply to it.

One of the simplest examples of a category (which is a very important concept in topology) is that of groupoid,
defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of
mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics
where they can be used to describe vector spaces. Categories were first introduced by Samuel Eilenberg and
Saunders Mac Lane in 1942—45, in connection with algebraic topology.

Category theory has several faces known not just to specialists, but to other mathematicians. A term dating from the
1940s, "general abstract nonsense", refers to its high level of abstraction, compared to more classical branches of
mathematics. Homological algebra is category theory in its aspect of organising and suggesting manipulations in
abstract algebra. Diagram chasing is a visual method of arguing with abstract "arrows" joined in diagrams. Note that
arrows between categories are called functors, subject to specific defining commutativity conditions; moreover,
categorical diagrams and sequences can be defined as functors (viz. Mitchell, 1965). An arrow between two functors
is a natural transformation when it is subject to certain naturality or commutativity conditions. Functors and natural
transformations ('naturality') are the key concepts in category theory. Topos theory is a form of abstract sheaf
theory, with geometric origins, and leads to ideas such as pointless topology. A topos can also be considered as a
specific type of category with two additional topos axioms.

Background

The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related
mathematical structures by relating them to the structure-preserving functions between them. A systematic study of
category theory then allows us to prove general results about any of these types of mathematical structures from the
axioms of a category.

Consider the following example. The class Grp of groups consists of all objects having a "group structure". One can
proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is
immediately proved from the axioms that the identity element of a group is unique.

Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory
emphasizes the morphisms — the structure-preserving mappings — between these objects; by studying these
morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the

Category theory 198

group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a precise
sense — it is a "process" taking one group to another, in a way that carries along information about the structure of
the first group into the second group. The study of group homomorphisms then provides a tool for studying general
properties of groups and consequences of the group axioms.

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps
(morphisms) between topological spaces in topology (the associated category is called Top), and the study of smooth
functions (morphisms) in manifold theory.

If one axiomatizes relations instead of functions, one obtains the theory of allegories.

Functors

Abstracting again, a category is itself a type of mathematical structure, so we can look for "processes" which
preserve this structure in some sense; such a process is called a functor. A functor associates to every object of one
category an object of another category, and to every morphism in the first category a morphism in the second.

In fact, what we have done is define a category of categories and functors — the objects are categories, and the
morphisms (between categories) are functors.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms
between them; we are studying the relationships between various classes of mathematical structures. This is a
fundamental idea, which first surfaced in algebraic topology. Difficult topological questions can be translated into
algebraic questions which are often easier to solve. Basic constructions, such as the fundamental group or
fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of
groupoids in this way, and the concept is pervasive in algebra and its applications.

Natural transformation

Abstracting yet again, constructions are often "naturally related" — a vague notion, at first sight. This leads to the
clarifying concept of natural transformation, a way to "map" one functor to another. Many important constructions in
mathematics can be studied in this context. "Naturality" is a principle, like general covariance in physics, that cuts
deeper than is initially apparent.

Historical notes

In 1942—45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations
as part of their work in topology, especially algebraic topology. Their work was an important part of the transition
from intuitive and geometric homology to axiomatic homology theory. Eilenberg and Mac Lane later wrote that their
goal was to understand natural transformations; in order to do that, functors had to be defined, which required
categories.

Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in
Poland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in some
sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes;
Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes
preserving that structure. In order to achieve this understanding, Eilenberg and Mac Lane proposed an axiomatic
formalization of the relation between structures and the processes preserving them.

The subsequent development of category theory was powered first by the computational needs of homological
algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either
axiomatic set theory or the Russell-Whitehead view of united foundations. General category theory, an extension of
universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later; it is
now applied throughout mathematics.

Category theory 199

Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a
foundation of mathematics. These foundational applications of category theory have been worked out in fair detail as
a basis for, and justification of, constructive mathematics. More recent efforts to introduce undergraduates to
categories as a foundation for mathematics include Lawvere and Rosebrugh (2003) and Lawvere and Schanuel
(1997).

Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in
functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description
of a lambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in
common (in some abstract sense).

Categories, objects, and morphisms

A category C consists of the following three mathematical entities:

• A class ob(C), whose elements are called objects;

• A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism/ has a unique source
object a and target object b.

The expression/: a — > b, would be verbally stated as "/is a morphism from a to b".

The expression hom(«, b) — alternatively expressed as hom(a, b), or hom (a, b), or mor(a, b), or C(a, b) —

denotes the hom-class of all morphisms from a to b.

• A binary operation o , called composition of morphisms, such that for any three objects a, b, and c, we have

hom(a, b) x hom(/?, c) — > hom(a, c). The composition off: a — > b and g: b —> c is written as g o for gf,
governed by two axioms:

• Associativity: Iff: a — > b, g : b — > c and h: c — > d then h o (g o f\ = (hog)of, and

• Identity: For every object x, there exists a morphism 1 : x — > x called the identity morphism for x, such that for
every morphism/: a — > ft, we have \\>o f = f = f o\ a .

From these axioms, it can be proved that there is exactly one identity morphism for every object. Some authors
deviate from the definition just given by identifying each object with its identity morphism.

Relations among morphisms (such as/g = h) are often depicted using commutative diagrams, with "points" (corners)
representing objects and "arrows" representing morphisms.

Properties of morphisms

Morphisms can have any of the following properties. A morphism/: a — > b is a:

monomorphism (or monic) if fog =fog implies g = g for all morphisms g , g : x — > a.

epimorphism (or epic) if g.of- g of implies g = g for all morphisms g , g : b — > x.

bimorphism if/ is both epic and monic.

isomorphism if there exists a morphism g : b — > a such that/og =1 and gof= 1 .

endomorphism if a - b. end(a) denotes the class of endomorphisms of a.

automorphism if/ is both an endomorphism and an isomorphism, aut(a) denotes the class of automorphisms of a.

retraction if a right inverse off exists, i.e. if there exists a morphism g : b — > a with/g = 1 .

section if a left inverse off exists, i.e. if there exists a morphism g : b — > a with gf= 1 .

Every retraction is an epimorphism, and every section is a monomorphism. Hence the following three statements are
equivalent:

• /is a monomorphism and a retraction;

• /is an epimorphism and a section;

• /is an isomorphism.

Category theory 200

Functors

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of
all (small) categories.

A (covariant) functor F from a category C to a category D, written F:C — > D, consists of:

• for each object x in C, an object F(x) in D; and

• for each morphism/: x — > y in C, a morphism F(f) : F(x) — > F(y),

such that the following two properties hold:

• For every object x in C, F(l ) = 1 „, •

J x F(x)

• For all morphisms/: x — > y and g : y — » z, .Ffg o /) = Fig) o F(f).

A contravariant functor F: C — > D, is like a covariant functor, except that it "turns morphisms around" ("reverses all
the arrows"). More specifically, every morphism/: x — > y in C must be assigned to a morphism F(/) : F(y) — > F(x) in
Z). In other words, a contravariant functor is a covariant functor from the opposite category C op to D.

Natural transformations and isomorphisms

A natural transformation is a relation between two functors. Functors often describe "natural constructions" and
natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two
quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two
functors.

If F and G are (covariant) functors between the categories C and D, then a natural transformation r| from F to G
associates to every object X in C a morphism r\ : F{X) — > G(X) in D such that for every morphism/ : X — > Y in C,
we have T) o F(/) = G(f) o r| ; this means that the following diagram is commutative:

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such
that r| is an isomorphism for every object X in C.

Universal constructions, limits, and colimits

Using the language of category theory, many areas of mathematical study can be cast into appropriate categories,
such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are
"special" in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category,
objects are considered to be atomic, i.e., we do not know whether an object A is a set, a topology, or any other
abstract concept — hence, the challenge is to define special objects without referring to the internal structure of those
objects. But how can we define the empty set without referring to elements, or the product topology without
referring to open sets?

The solution is to characterize these objects in terms of their relations to other objects, as given by the morphisms of
the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest.

Category theory 201

Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central
concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a
colimit.

Equivalent categories

It is a natural question to ask: under which conditions can two categories be considered to be "essentially the same",
in the sense that theorems about one category can readily be transformed into theorems about the other category?
The major tool one employs to describe such a situation is called equivalence of categories, which is given by
appropriate functors between two categories. Categorical equivalence has found numerous applications in
mathematics.

Further concepts and results

The definitions of categories and functors provide only the very basics of categorical algebra; additional important
topics are listed below. Although there are strong interrelations between all of these topics, the given order can be
considered as a guideline for further reading.

• The functor category D has as objects the functors from C to D and as morphisms the natural transformations of
such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes
representable functors in functor categories.

• Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by
"reversing all the arrows". If one statement is true in a category C then its dual will be true in the dual category
C° p . This duality, which is transparent at the level of category theory, is often obscured in applications and can
lead to surprising relationships.

• Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction.
Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be
seen as a more abstract and powerful view on universal properties.

Higher-dimensional categories

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can
be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two
objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably
generalize this by considering "higher-dimensional processes".

For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e., processes which
allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally
and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this
context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of
morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider
a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of
2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up
to" an isomorphism.

This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of
m-category corresponding to the ordinal number m.

Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept
introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, A Tale of
n-categories' (1996).

Category theory 202

Notes

[1] Categories for the Working Mathematician, 2nd Edition, p 18: "As Eilenberg-Mac Lane first observed, 'category' has been defined in order to

be able to define 'functor' and 'functor' has been defined in order to be able to define 'natural transformation' ".

[2] http://planetphysics.org/encyclopedia/FundamentalGroupoidFunctor.html

[3] Some authors compose in the opposite order, writing/g or J O g for g O J . Computer scientists using category theory very commonly

write fig for g O J
[4] Note that a morphism that is both epic and monic is not necessarily an isomorphism! An elementary counterexample: in the category

consisting of two objects A and B, the identity morphisms, and a single morphism/' from A to B, /is both epic and monic but is not an

isomorphism.

[5] http://math.ucr.edu/home/baez/week73.html

References

• Adamek, Jiff; Herrlich, Horst; Strecker, George E. (1990). Abstract and concrete categories (http://katmat.math.
uni-bremen.de/acc/acc.htm). John Wiley & Sons. ISBN 0-471-60922-6.

• Awodey, Steve (2006). Category Theory (Oxford Logic Guides 49). Oxford University Press. 2nd edition, 2010.

• Barr, Michael; Wells, Charles (1999). "Category Theory Lecture Notes" (http://folli.loria.fr/cds/1999/library/
pdf/barrwells.pdf). Retrieved 11 December 2009-12-11. Based on their book Category Theory for Computing
Science, Centre de recherches mathematiques CRM (http://crm.umontreal.ca/pub/Ventes/desc/PM023.html),
1999.

Barr, Michael; Wells, Charles (2002). Toposes, triples and theories (http://www.cwru.edu/artsci/math/wells/
pub/ttt.html). Revised and corrected web publication of Toposes, Triples and Theories. Grundlehren der
mathematischen Wissenschaften. Springer- Verlag. 1983.

Borceux, Francis (1994). Handbook of categorical algebra (Encyclopedia of Mathematics and its Applications
50-52). Cambridge Univ. Press.

I. Bucur, A. Deleanu (1968). Introduction to the theory of categories and functors, Wiley.
Freyd, Peter J. (1964). Abelian Categories (http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html). New
York: Harper and Row.

Freyd, Peter J.; Scedrov, Andre (1990). Categories, allegories. North Holland Mathematical Library 39. North
Holland.

Goldblatt, R. (1984). Topoi: The Categorial Analysis of Logic (http://dlxs2.library.cornell.edU/cgi/t/text/
text-idx?c=math;cc=math;view=toc;subview=short;idno=Gold010).

Hatcher, William S. (1982). "8". In Pergamon. The Logical Foundations of Mathematics (2nd ed.).
Herrlich, Horst; Strecker, George E. (2007), Category Theory (3rd ed.), Heldermann Verlag Berlin,
ISBN 978-3-88538-001-6.

Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften
332, Springer (2000)

Lawvere, William; Rosebrugh, Robert (2003). Sets for mathematics. Cambridge University Press.
Lawvere, William; Schanuel, Steve (1997). Conceptual mathematics: a first introduction to categories.
Cambridge University Press.

Leinster, Tom (2004). Higher operads, higher categories (http://www.maths.gla.ac.uk/~tl/book.html).
London Math. Society Lecture Note Series 298. Cambridge University Press.

Lurie, Jacob (2009). Higher topos theory. Annals of Mathematics Studies 170. Princeton University Press.
arXiv:math.CT/0608040.

Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd
ed.). Springer- Verlag. ISBN 0-387-98403-8.

Mac Lane, Saunders; Birkhoff, Garrett (1999) [1967]. Algebra (2nd ed.). Chelsea. ISBN 0-8218-1646-2.
Martini, A.; Ehrig, H.; Nunes, D. (1996). "Elements of basic category theory" (http://citeseer.ist.psu.edu/
martini96element.html). Technical Report (Technical University Berlin) 96 (5).

Category theory 203

• May, Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. ISBN 0-226-51 183-9.

• Pedicchio, Maria Cristina; Tholen, Walter (2004). Categorical foundations . Encyclopedia of Mathematics and its
Applications 97. Cambridge University Press.

• Pierce, Benjamin (1991). Basic Category Theory for Computer Scientists. MIT Press.

• Schalk, A.; Simmons, H. (2005). An introduction to Category Theory in four easy movements (http://www.es.
man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf). Notes for a course offered as part of the MSc. in
Mathematical Logic, Manchester University.

• Simpson, Carlos. Homotopy theory of higher categories. arXiv: 1001 .407 1 ., draft of a book.

• Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press.

• Turi, Daniele (1996—2001). "Category Theory Lecture Notes" (http://www.dcs.ed.ac.uk/home/dt/CT/
categories.pdf). Retrieved 1 1 December 2009. Based on Mac Lane (1998).

External links

• Chris Hillman, A Categorical Primer (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.3264&
rep=repl&type=pdf), formal introduction to category theory.

• J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats (http://katmat.math.
uni-bremen.de/acc/acc.pdf)

• Stanford Encyclopedia of Philosophy: " Category Theory (http://plato.stanford.edu/entries/category-theory/)"
— by Jean-Pierre Marquis. Extensive bibliography.

• List of academic conferences on category theory (http://www.mta.ca/~cat-dist/)

• Baez, John, 1996," The Tale of n-categories. (http://math.ucr.edu/home/baez/week73.html)" An informal
introduction to higher order categories.

• The catsters (http://www.youtube.com/user/TheCatsters) , a Youtube channel about category theory.

• Category Theory (http://planetmath.org/?op=getobj&from=objects&id=5622) on PlanetMath

• Video archive (http://categorieslogicphysics.wikidot.com/events) of recorded talks relevant to categories, logic
and the foundations of physics.

• Interactive Web page (http://www.j-paine.org/cgi-bin/webcats/webcats.php) which generates examples of
categorical constructions in the category of finite sets.

Higher-dimensional algebra 204

Higher-dimensional algebra

This article is about higher-dimensional algebra and supercategories in generalized category theory,
super-category theory, and also its extensions in nonabelian algebraic topology and metamathematics.

T21
Supercategories were first introduced in 1970, and were subsequently developed for applications in theoretical

physics (especially quantum field theory and topological quantum field theory) and mathematical biology or

mathematical biophysics.

Double groupoids, fundamental groupoids, 2-categories, categorical QFTs and
TQFTs

In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two

mi
dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or

morphisms.

Double groupoids are often used to capture information about geometrical objects such as higher-dimensional
manifolds (or « -dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an
«-dimensional Euclidean space, but whose global structure may be non-Euclidean. A first step towards defining
higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more "geometric'
concept of double category . Other pathways in HDA involve: bicategories, homomorphisms of bicategories,

variable categories (aka, indexed, or parametrized categories), topoi, effective descent, enriched and internal
categories, as well as quantum categories and quantum double groupoids. In the latter case, by

considering fundamental groupoids defined via a 2-functor allows one to think about the physically interesting case
of quantum fundamental groupoids (QFGs) in terms of the bicategory Span(Groupoids), and then constructing
2-Hilbert spaces and 2-linear maps for manifolds and cobordisms. At the next step, one obtains cobordisms with

corners via natural transformations of such 2-functors. A claim was then made that, with the gauge group SU(2),

ri3i
"the extended TQFT, or ETQFT, gives a theory equivalent to the Ponzano-Regge model of quantum gravity";

similarly, the Turaev-Viro model would be then obtained with representations of SU_q(2). Therefore, according to

the construction proposed by Jeffrey Morton, one can describe the state space of a gauge theory — or many kinds of

quantum field theories (QFTs) and local quantum physics, in terms of the transformation groupoids given by

symmetries, as for example in the case of a gauge theory, by the gauge transformations acting on states that are, in

this case, connections. In the case of symmetries related to quantum groups, one would obtain structures that are

ri4i
representation categories of quantum groupoids, instead of the 2-vector spaces that are representation categories

of groupoids.

Double categories, Category of categories and Supercategories

A higher level concept is thus defined as a category of categories, or super-category, which generalises to higher
dimensions the notion of category — regarded as any structure which is an interpretation of Lawvere's axioms of the
elementary theory of abstract categories (ETAC). Thus, a supercategory and also a super-category,

ri9i

can be regarded as natural extensions of the concepts of meta-category, multicategory, and multi-graph, k-partite
graph, or colored graph (see a color figure, and also its definition in graph theory).

Double groupoids were first introduced by Ronald Brown in 1976, in ref. and were further developed towards

r211 T221 r231 [241

applications in nonabelian algebraic topology. A related, 'dual' concept is that of a double algebroid,

and the more general concept of R-algebroid.

Higher-dimensional algebra 205

Nonabelian algebraic topology

Many of the higher dimensional algebraic structures are noncommutative and, therefore, their study is a very

significant part of nonabelian category theory, and also of Nonabelian Algebraic Topology (NAAT) which

[271
generalises to higher dimensions ideas coming from the fundamental group. Such algebraic structures in

dimensions greater than 1 develop the nonabelian character of the fundamental group, and they are in a precise sense

no] [291

'more nonabelian than the groups' . These noncommutative, or more specifically, nonabelian structures

reflect more accurately the geometrical complications of higher dimensions than the known homology and homotopy
groups commonly encountered in classical algebraic topology. An important part of nonabelian algebraic topology is
concerned with the properties and applications of homotopy groupoids and filtered spaces. Noncommutative double
groupoids and double algebroids are only the first examples of such higher dimensional structures that are
nonabelian. The new methods of Nonabelian Algebraic Topology (NAAT) "can be applied to determine homotopy
invariants of spaces, and homotopy classification of maps, in cases which include some classical results, and allow
results not available by classical methods". Cubical omega-groupoids, higher homotopy groupoids, crossed
modules, crossed complexes and Galois groupoids are key concepts in developing applications related to homotopy
of filtered spaces, higher dimensional space structures, the construction of the fundamental groupoid of a topos E in

the general theory of topoi, and also in their physical applications in nonabelian quantum theories, and recent

[311
developments in quantum gravity, as well as categorical and topological dynamics. Further examples of such

applications include the generalisations of noncommutative geometry formalizations of the noncommutative

standard models via fundamental double groupoids and spacetime structures even more general than topoi or the

lower-dimensional noncommutative spacetimes encountered in several topological quantum field theories and

noncommutative geometry theories of quantum gravity.

A fundamental result in NAAT is the generalised, higher homotopy van Kampen theorem proven by R. Brown
which states that "the homotopy type of a topological space can be computed by a suitable colimit or homotopy

colimit over homotopy types of its pieces" . A related example is that of van Kampen theorems for categories of

T321
covering morphisms in lextensive categories. Other reports of generalisations of the van Kampen theorem include

T331
statements for 2-categories and a topos of topoi [34]. Important results in HDA are also the extensions of the

Galois theory in categories and variable categories, or indexedAparametrized' categories. The Joyal-Tierney

T371
representation theorem for topoi is also a generalisation of the Galois theory. Thus, indexing by bicategories in

T3R1

the sense of Benabou one also includes here the Joyal-Tierney theory.

Notes

[I] Roger Bishop Jones. 2008. The Category of Categories http://www.rbjones.com/rbjpub/pp/doc/t018.pdf
[2] Supercategory theory @ PlanetMath (http://planetmath.org/encyclopedia/Supercategories3.html)

[3] http://planetphysics.org/encyclopedia/MathematicalBiologyAndTheoreticalBiophysics.html

[4] Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules,". Cahiers Top. Geom. Diff. 17: 343—362.

[5] Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules" (http://www.bangor.ac.uk/~mas010/pdffiles/

brown-spencerCTGDC_1976__17_4_343_0.pdf). Cah. Top. Geom. Diff 11: 343-362. .
[6] http://www.math.uchicago.edU/~fiore/l/fiorefolding.pdf
[7] Brown, R.; Loday, J.-L. (1987). "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces". Proceedings of the London

Mathematical Society 3(54): 176-192. doi: 10. 1006/aima. 1998. 1724.
[8] Batanin, M.A. (1998). "Monoidal Globular Categories As a Natural Environment for the Theory of Weak n-Categories". Advances in

Mathematics 136 (1): 39-103. doi:10.1006/aima.l998.1724.
[9] http://planetmath.org/encyclopedia/QuantumCategory.html Quantum Categories of Quantum Groupoids
[10] http://planetmath.org/encyclopedia/AssociativityIsomorphism.html Rigid Monoidal Categories

[II] http://theoreticalatlas.wordpress.com/2009/03/18/a-note-on-quantum-groupoids/

[12] http://theoreticalatlas.wordpress.com/2009/03/18/a-note-on-quantum-groupoids/March 18, 2009. A Note on Quantum Groupoids,
posted by Jeffrey Morton under C*-algebras, deformation theory, groupoids, noncommutative geometry, quantization

[13] http://theoreticalatlas.wordpress.com/2009/03/18/a-note-on-quantum-groupoids/March 18, 2009. A Note on Quantum Groupoids,
posted by Jeffrey Morton under C*-algebras, deformation theory, groupoids, noncommutative geometry, quantization

[14] http://planetmath.org/encyclopedia/QuantumCategory.html Quantum Categories of Quantum Groupoids

Higher-dimensional algebra 206

[15] Lawvere, F. W., 1964, "An Elementary Theory of the Category of Sets, Proceedings of the National Academy of Sciences U.S.A., 52,

1506—1511. http://myyn.org/rn/article/william-francis-lawvere/
[16] Lawvere, F. W.: 1966, The Category of Categories as a Foundation for Mathematics., in Proc. Conf. Categorical Algebra —La Jolla.,

Eilenberg, S. et al., eds. Springer- Verlag: Berlin, Heidelberg and New York., pp. 1—20. http://myyn.org/rn/article/william-francis-lawvere/
[17] http://planetphysics.org/?op=getobj&from=objects&id=420

[18] Lawvere, F. W., 1969b, "Adjointness in Foundations, Dialectica, 23, 281—295. http://myyn.org/rn/article/william-francis-lawvere/
[19] http://planetphysics.org/encyclopedia/AxiomsOfMetacategoriesAndSupercategories.html
[20] Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules" (http://www.bangor.ac.uk/~mas010/pdffiles/

brown-spencerCTGDC_1976__17_4_343_0.pdf). Cah. Top. Ge'om. Diff 17: 343-362. .
[2 1 ] http://planetphysics. org/encyclopedia/NAAT. html

[22] Non-Abelian Algebraic Topology book (http://www.bangor.ac.uk/~mas010/nonab-a-t.html)
[23] Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces (http://planetphysics.org/?op=getobj&from=books&

id=249)
[24] Brown, R.; et al. (2009) (in press). Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces (http://www.bangor.ac.

uk/~mas010/pdffiles/rbrsbookb-e040609.pdf). .
[25] *Brown, R.; Higgins, P.J.; Sivera, R. (2008). Non-Abelian Algebraic Topology (http://www.bangor.ac.uk/~mas010/nonab-a-t.html). 1. .

( Downloadable PDF (http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf))
[26] http://www.ems-ph.org/pdf/catalog.pdf Ronald Brown, Philip Higgins, Rafael Sivera, Nonabelian Algebraic Topology: Filtered spaces,

crossed complexes, cubical homotopy groupoids, in Tracts in Mathematics vol. 15 (2010), European Mathematical Society, 670 pages, ISBN

978-3-03719-083-8
[27] http://arxiv.org/abs/math/0407275 Nonabelian Algebraic Topology by Ronald Brown. 15 Jul 2004
[28] http://golem.ph.utexas.edu/category/2009/06/nonabelian_algebraic_topology.html Nonabelian Algebraic Topology posted by John

Baez
[29] *Brown, R.; Higgins, P.J.; Sivera, R. (2008). Non-Abelian Algebraic Topology (http://www.bangor.ac.uk/~mas010/nonab-a-t.html). 1. .

( Downloadable PDF (http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf))
[30] http://planetphysics.org/?op=getobj&from=books&id=374 Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes and

Cubical Homotopy groupoids, by Ronald Brown, Bangor University, UK, Philip J. Higgins, Durham University, UK Rafael Sivera, University

of Valencia, Spain
[31] http://www.springerlink.com/content/92rl3230n3381746/ A Conceptual Construction of Complexity Levels Theory in Spacetime

Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems by R. Brown et al., Axiomathes,

Volume 17, Numbers 3-4, 409-493, DOI: 10.1007/sl0516-007-9010-3
[32] Ronald Brown and George Janelidze, van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl.

Algebra. 119:255-263, (1997)
[33] http://webcache.googleusercontent.com/search?q=cache:nJq91pZj-3gJ: www. maths. usyd.edu. au/u/stevel/papers/vkt.ps.gz+

Kampen+theorems+for+toposes&cd=l&hl=en&ct=clnk&gl=us Marta Bunge and Stephen Lack. Van Kampen theorems for 2-categories

and toposes
[34] http://www. maths. usyd.edu. au/u/stevel/papers/vkt.ps.gz

[35] http://www.springerlink.com/content/gugl4ull41214743/George Janelidze, Pure Galois theory in categories, J. Alg. 132:270—286, 199
[36] http://www.springerlink.com/content/gugl4ul 141214743/Galois theory in variable categories., by George Janelidze, Dietmar

Schumacher and Ross Street, in APPLIED CATEGORICAL STRUCTURES, Volume 1, Number 1, 103-110, doi:10.1007/BF00872989
[37] Joyal, Andres; Tierney, Myles (1984). An extension of the Galois theory of Grothendieck. 309. American Mathematical Society.

ISBN 0821823124.
[38] MSC(1991): 18D30,11R32,18D35,18D05

Further reading

• Brown, R.; Higgins, P.J.; Sivera, R. (2008). Non-Abelian Algebraic Topology (http://www.bangor.ac.uk/
~mas010/nonab-a-t.html). 1. ( Downloadable PDF (http://www.bangor.ac.uk/~mas010/nonab-t/
partI010604.pdf))

• Brown, R.; Spencer, C.B. (1976). "Double groupoids and crossed modules,". Cahiers Top. Geom. Diff. 17:
343-362.

• Brown, R.; Mosa, G.H. (1999). "Double categories, thin structures and connections". Theory and Applications of
Categories 5: 163—175.

• Brown, R. (2002). Categorical Structures for Descent and Galois Theory. Fields Institute.

• Brown, R. (1987). "From groups to groupoids: a brief survey" (http://www.bangor.ac.Uk/r.brown/
groupoidsurvey.pdf). Bulletin of the London Mathematical Society 19: 1 13— 134. doi:10.1 1 12/blms/19.2.1 13. This

Higher-dimensional algebra 207

give some of the history of groupoids, namely the origins in work of Heinrich Brandt on quadratic forms, and an
indication of later work up to 1987, with 160 references.

• Brown, R.. "Higher dimensional group theory" (http://www.bangor.ac.Uk/r.brown/hdaweb2.htm).. A web
article with lots of references explaining how the groupoid concept has to led to notions of higher dimensional
groupoids, not available in group theory, with applications in homotopy theory and in group cohomology.

• Brown, R.; Higgins, P.J. (1981). "On the algebra of cubes". Journal of Pure and Applied Algebra 21: 233—260.
doi:10.1016/0022-4049(81)90018-9.

• Mackenzie, K.C.H. (2005). General theory of Lie groupoids and Lie algebroids (http://www.shef.ac.uk/
-pmlkchm/gt.html). Cambridge University Press.

• R., Brown (2006). Topology and groupoids (http://www.bangor.ac.Uk/r.brown/topgpds.html). Booksurge.
Revised and extended edition of a book previously published in 1968 and 1988. E-version available at
PlanetPhysics.org (http://planetphysics.org/?op=getobj&from=lec&id=177) and Bangor.ac.uk (http://www.
bangor. ac . uk/~masO 1 0/topgpds . html)

• Borceux, F.; Janelidze, G. (2001). Galois theories (http://www.cup.cam.ac.uk/catalogue/catalogue.
asp?isbn=978052 1803090). Cambridge University Press. Shows how generalisations of Galois theory lead to
Galois groupoids.

• Baez, J.; Dolan, J. (1998). "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes".
Advances in Mathematics 135: 145-206. doi:10.1006/aima. 1997. 1695.

• Baianu, I.C. (1970). "Organismic Supercategories: II. On Multistable Systems". Bulletin of Mathematical
Biophysics (http://www.springerlink.com/content/x513p402w52wll28/) 32: 539—561.
doi:10.1007/BF02476770. PMID 4327361.

• Baianu, I.C; Marinescu, M. (1974). "On A Functorial Construction of (M, 7?)-Systems". Revue Roumaine de
Mathematiques Pures et Appliquees 19: 388—391.

• Baianu, I.C. (1987). "Computer Models and Automata Theory in Biology and Medicine" (http://cogprints.org/
3687/). In M. Witten. Mathematical Models in Medicine (http://www.amazon.ca/

Mathematical-Models-Medicine-Diseases-Epidemics/dp/0080346928). 7. Pergamon Press, pp. 1513—1577.
CERN Preprint No. EXT-2004-072..

• "Higher dimensional Homotopy @ PlanetPhysics" (http://planetphysics.org/encyclopedia/
HigherDimensionalHomotopy.html).

• George Janelidze, Pure Galois theory in categories, J. Alg. 132:270—286, 1990. (http://www.springerlink.com/
content/gugl4ul 141214743/)

• Galois theory in variable categories., by George Janelidze, Dietmar Schumacher and Ross Street, in APPLIED
CATEGORICAL STRUCTURESVolume 1, Number 1, 103-110, DOI: 10.1007/BF00872989 (http://www.
springerlink. com/content/gug 1 4u 1141214743/).

Higher category theory 208

Higher category theory

Higher category theory is the part of category theory at a higher order, which means that some equalities are
replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.

Strict higher categories

N-categories are defined inductively using the enriched category theory: 0-categories are sets, and (n+l)-categories
are categories enriched over the monoidal category of n-categories (with the monoidal structure given by finite
products). This construction is well defined, as shown in the article on n-categories. This concept introduces
higher arrows, higher compositions and higher identities, which must well behave together. For example, the
category of small categories is in fact a 2-category, with natural transformations as second degree arrows. However
this concept is too strict for some purposes (for example, homotopy theory), where "weak" structures arise in the
form of higher categories.

Weak higher categories

In weak n-categories, the associativity and identity conditions are no longer strict (that is, they are not given by
equalities), but rather are satisfied up to an isomorphism of the next level. An example in topology is the
composition of paths, which is associative only up to homotopy. These isomorphisms must well behave between
hom-sets and expressing this is the difficulty in the definition of weak n-categories. Weak 2-categories, also called
bicategories, were the first to be defined explicitly. A particularity of these is that a bicategory with one object is
exactly a monoidal category, so that bicategories can be said to be "monoidal categories with many objects." Weak
3-categories, also called tricategories, and higher-level generalizations are increasingly harder to define explicitly.
Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new
object of study in category theory.

Quasicategories

Weak Kan complexes, or quasi-categories, are semisimplicial complexes satisfying a weak version of the Kan
condition. Joyal showed that they are a good foundation for higher category theory. Recently the theory has been
systematized further by Jacob Lurie who simply call them infinity categories, though the latter term is also a generic
term for all models of (infinity ,k) categories for any k.

Simplicially enriched category

Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However,
when we look at them as a model for (infinity, l)-categories, then many categorical notions, say limits do not agree
with the corresponding notions in the sense of enriched categories. The same for other enriched models like
topologically enriched categories.

Higher category theory 209

Topologically enriched categories

Topologically enriched categories (sometimes simply topological categories) are categories enriched over some
convenient category of topological spaces, e.g. the category of compactly generated Hausdorff topological spaces.

Segal categories

These are models of higl
of Graeme Segal in 1974.

These are models of higher categories introduced by Hirschowitz and Simpson in 1988 , partly inspired by results

References

[1] Leinster, pp 18-19

[2] Baez, p 6

[3] Andre Hirschowitz, Carlos Simpson (1998), Descente pour les n-champs (Descent for n-stacks)

• John C. Baez; James Dolan (1998). Categorification. arXiv:math/9802029.

• Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. arXiv:math.CT/0305049.
ISBN 0521532159.

• Carlos Simpson, Homotopy theory of higher categories, draft of a book arXiv: 100 1.4071 (alternative URL with
hyperTeX-ed crosslinks: pdf (http://hal.archives-ouvertes.fr/docs/00/44/98/26/PDF/main.pdf))

• Jacob Lurie, Higher topos theory, arXiv:math.CT/0608040, published version: pdf (http://www.math. harvard.
edu/~lurie/papers/highertopoi.pdf)

• nlab (http://ncatlab.org/nlab/show/HomePage), the collective and open wiki notebook project on higher
category theory and applications in physics, mathematics and philosophy

• Joyal's Catlab (http://ncatlab.org/joyalscatlab/show/HomePage), a wiki dedicated to polished expositions of
categorical and higher categorical mathematics with proofs

External links

• John Baez Tale of n-Categories (http://math.ucr.edu/home/baez/week73.html)

• The n-Category Cafe (http://golem.ph.utexas.edu/category/) - a group blog devoted to higher category theory.

Algebraic topology 210

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces.
The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually
most classify up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic
problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any
subgroup of a free group is again a free group.

The method of algebraic invariants

An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed
from simpler ones (the modern standard tool for such construction is the CW-complex). The basic method now
applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to
groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or
more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements
about groups, which are often easier to prove.

Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and
through homology and cohomology groups. The fundamental groups give us basic information about the structure of
a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a
(finite) simplicial complex does have a finite presentation.

Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated.
Finitely generated abelian groups are completely classified and are particularly easy to work with.

Setting in category theory

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural
transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants
of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same
associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a
group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or,
much more deeply, existence) of mappings.

Results on homology

Several useful results follow immediately from working with finitely generated abelian groups. The free rank of the
n-th homology group of a simplicial complex is equal to the «-th Betti number, so one can use the homology groups
of a simplicial complex to calculate its Euler-Poincare characteristic. As another example, the top-dimensional
integral homology group of a closed manifold detects orientability: this group is isomorphic to either the integers or
0, according as the manifold is orientable or not. Thus, a great deal of topological information is encoded in the
homology of a given topological space.

Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure
of smooth manifolds via de Rham cohomology, or Cech or sheaf cohomology to investigate the solvability of
differential equations defined on the manifold in question. De Rham showed that all of these approaches were
interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were
the same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s, when
Eilenberg and Steenrod generalized this approach. They defined homology and cohomology as functors equipped

Algebraic topology 211

with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism
of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that
such an axiomatization uniquely characterized the theory.

A new approach uses a functor from filtered spaces to crossed complexes defined directly and homotopically using
relative homotopy groups; a higher homotopy van Kampen theorem proved for this functor enables basic results in
algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular
homology or simplicial approximation. This approach is also called nonabelian algebraic topology, and generalises
to higher dimensions ideas coming from the fundamental group.

Applications of algebraic topology

Classic applications of algebraic topology include:

• The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.

• The «-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n - 2, this is
sometimes called the "hairy ball theorem".)

• The Borsuk— Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one
pair of antipodal points.

• Any subgroup of a free group is free. This result is quite interesting, because the statement is purely algebraic yet
the simplest proof is topological. Namely, any free group G may be realized as the fundamental group of a graph
X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some
covering space 7 of X; but every such Y is again a graph. Therefore its fundamental group H is free.

On the other hand this type of application is also handled by the use of covering morphisms of groupoids, and that
technique has yielded subgroup theorems not yet proved by methods of algebraic topology (see the book by Higgins
listed under groupoids).

• Topological combinatorics

Notable algebraic topologists

Frank Adams

Karol Borsuk

Luitzen Egbertus Jan Brouwer

William Browder

Ronald Brown (mathematician)

Henri Cartan

Samuel Eilenberg

Hans Freudenthal

Peter Freyd

Alexander Grothendieck

Friedrich Hirzebruch

Heinz Hopf

Michael J. Hopkins

Witold Hurewicz

Egbert van Kampen

Daniel Kan

Hermann Kunneth

Solomon Lefschetz

Jean Leray

Algebraic topology 212

Saunders Mac Lane
Mark Mahowald
J. Peter May
John Coleman Moore
Jack Morava
Goro Nishida
Sergei Novikov
Lev Pontryagin
Mikhail Postnikov
Daniel Quillen
Jean-Pierre Serre
Stephen Smale
Norman Steenrod
Dennis Sullivan
Rene Thom
Hiroshi Toda
Leopold Vietoris
Hassler Whitney
J. H. C. Whitehead

Important theorems in algebraic topology

Borsuk-Ulam theorem
Brouwer fixed point theorem
Cellular approximation theorem
Eilenberg— Zilber theorem
Freudenthal suspension theorem
Hurewicz theorem
Kunneth theorem
Poincare duality theorem
Universal coefficient theorem
Van Kampen's theorem

Generalized van Kampen's theorems

121 131
Higher homotopy, generalized van Kampen's theorem

Whitehead's theorem

Notes

[i] http://pianetphysics.org/encyciopedia/GeneraiizedvanKampenTheoremsHDGVKT.htmi#BHKP

[2] R. Brown, K. A. Hardie, K. H. Kamps and T. Porter (2002), "A homotopy double groupoid of a Hausdorff space", Theory and Applications of

Categories 10: 71-93.
[3] R. Brown, K. H. Kamps and T. Porter (2005), Theory and Applications of Categories 14 (9): 200-220.

References

• Bredon, Glen E. (1993), Topology and Geometry (http://books.google.com/?id=G74V6UzL_PUC&

printsec=frontcover&dq=bredon+topology+and+geometry), Graduate Texts in Mathematics 139, Springer,
ISBN 0-387-97926-3, retrieved 2008-04-01.

Algebraic topology 213

• Hatcher, Allen (2002), Algebraic Topology (http://www.math.cornell.edu/~hatcher/AT/ATpage.html),
Cambridge: Cambridge University Press, ISBN 0-521-79540-0. A modern, geometrically flavored introduction to
algebraic topology.

• Maunder, C. R. F. (1970), Algebraic Topology, London: Van Nostrand Reinhold, ISBN 0-486-69131-4.

• R. Brown and A. Razak, A van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984)
85-88.

• P. J. Higgins, Categories and groupoids (1971) Van Nostrand-Reinhold. (http://138.73.27.39/tac/reprints/
articles/7/tr7abs.html)

• Ronald Brown, Higher dimensional group theory (http://www.bangor.ac.Uk/r.brown/hdaweb2.html) (2007)
(Gives a broad view of higher dimensional van Kampen theorems involving multiple groupoids).

• E. R. van Kampen. On the connection between the fundamental groups of some related spaces. American Journal
of Mathematics, vol. 55 (1933), pp. 261-267.

• R. Brown, P.J. Higgins, and R. Sivera. Non-Abelian Algebraic Topology: filtered spaces, crossed complexes,
cubical higher homotopy groupoids (http://www.bangor.ac.uk/~mas010/nonab-a-t.html); European
Mathematical Society Tracts in Mathematics Vol. 15, 2010, downloadable PDF: (http://www.bangor.ac.uk/
~mas010/pdffiles/rbrsbookb-e090809.pdf)

• Van Kampen's theorem (http://planetmath.org/?op=getobj&from=objects&id=3947) on PlanetMath

• Van Kampen's theorem result (http://planetmath.org/?op=getobj&from=objects&id=5576) on
PlanetMath

• R. Brown, K. Hardie, H. Kamps, T. Porter: The homotopy double groupoid of a Hausdorff space., Theory Appl.
Categories, 10:71—93 (2002).

• Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem (http://www.math.uchicago.edu/~may/
VIGRE/VIGREREU2008.html) (Discusses generalized versions of van Kampen's theorem applied to
topological spaces and simplicial sets).

Further reading

• Allen Hatcher, Algebraic topology. (http://www.math.cornell.edu/~hatcher/AT/ATpage.html) (2002)
Cambridge University Press, Cambridge, xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0.

• May, J. P. (1999), A Concise Course in Algebraic Topology (http://www.math.uchicago.edu/~may/
CONCISE/ConciseRevised.pdf), U. Chicago Press, Chicago, retrieved 2008-09-27. (Section 2.7 provides a
category-theoretic presentation of the theorem as a colimit in the category of groupoids).

• Higher dimensional algebra

• Ronald Brown, Philip J. Higgins and Rafael Sivera. 2009. Higher dimensional, higher homotopy, generalized van
Kampen Theorem., in Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical higher
homotopy groupoids. 512 pp, (Preprint), (http://www.bangor.ac.uk/~mas010/pdffiles/rbrsbookb-e090809.

pdf)

• Ronald Brown, Topology and groupoids (http://www.bangor.ac.Uk/r.brown/topgpds.html) (2006) Booksurge
LLC ISBN 1-4196-2722-8.

Topological dynamics 214

Topological dynamics

In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative,
asymptotic properties of dynamical systems are studied from the viewpoint of general topology.

Scope

The central object of study in topological dynamics is a topological dynamical system, i.e. a topological space,
together with a continuous transformation, a continuous flow, or more generally, a semigroup of continuous
transformations of that space. The origins of topological dynamics lie in the study of asymptotical properties of
trajectories of systems of autonomous ordinary differential equations, in particular, the behavior of limit sets and
various manifestations of "repetitiveness" of the motion, such as periodic trajectories, recurrence and minimality,
stability, non-wandering points. George Birkhoff is considered to be the founder of the field. A structure theorem for
minimal distal flows proved by Hillel Furstenberg in the early 1960s inspired much work on classification of
minimal flows. A lot of research in the 1970s and 1980s was devoted to topological dynamics of one-dimensional
maps, in particular, piecewise linear self-maps of the interval and the circle.

Unlike the theory of smooth dynamical systems, where the main object of study is a smooth manifold with a
diffeomorphism or a smooth flow, phase spaces considered in topological dynamics are general metric spaces
(usually, compact). This necessitates development of entirely different techniques but allows extra degree of
flexibility even in the smooth setting, because invariant subsets of a manifold are frequently very complicated
topologically (cf limit cycle, strange attractor); additionally, shift spaces arising via symbolic representations can be
considered on an equal footing with more geometric actions. Topological dynamics has intimate connections with
ergodic theory of dynamical systems, and many fundamental concepts of the latter have topological analogues (cf
Kolmogorov— Sinai entropy and topological entropy).

References

• D.V.Anosov (2001), "Topological dynamics" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics,
Springer, ISBN 978-1556080104

• Topological dynamics at Scholarpedia, curated by Joseph Auslander.

• Robert Ellis, Lectures on topological dynamics. W. A. Benjamin, Inc., New York 1969

• Walter Gottschalk, Gustav Hedlund, Topological dynamics. American Mathematical Society Colloquium
Publications, Vol. 36. American Mathematical Society, Providence, R. I., 1955

• J. de Vries, Elements of topological dynamics. Mathematics and its Applications, 257. Kluwer Academic
Publishers Group, Dordrecht, 1993 ISBN 0-7923-2287-8

• Ethan Akin, The General Topology of Dynamical Systems, AMS Bookstore, 2010, ISBN 9780821849323

References

[1] http://eom.springer.de/T/t093030.htm

[2] http://www.scholarpedia.org/article/Topological_dynamics

Graph dynamical system 215

Graph dynamical system

In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking
place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate
their structural properties (e.g. the network connectivity) and the global dynamics that result.

The work on GDSs considers finite graphs and finite state spaces. As such, the research typically involves techniques
from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In
principle, one could define and study GDSs over an infinite graph (e.g. cellular automata over 2 fc ° r interacting
particle systems), as well as GDSs with infinite state space (e.g. ]^ as in coupled map lattices); see, e.g., Wu. In
the following everything is implicitly assumed to be finite unless stated otherwise.

Formal definition

A graph dynamical system is constructed from the following components:

• A finite graph Y with vertex set v[Y] = { 1,2, ... , n}. Depending on the context the graph can be directed or
undirected.

• Astatex for each vertex v of Y taken from a finite set K. The system state is the n-tuplex = (x , x , ... , x ), and
x[v] is the tuple consisting of the states associated to the vertices in the 1 -neighborhood of v in Y (in some fixed
order).

• A vertex function f for each vertex v. The vertex function maps the state of vertex v at time t to the vertex state
at time t + 1 based on the states associated to the 1 -neighborhood of v in Y.

• An update scheme specifying the mechanism by which the mapping of individual vertex states is carried out so
as to induce a discrete dynamical system with map F: a —> K .

The phase space associated to a dynamical system with map F: a —> a is the finite directed graph with vertex set
K" and directed edges (x, F(x)). The structure of the phase space is governed by the properties of the graph Y, the
vertex functions (f) ., and the update scheme. The research in this area seeks to infer phase space properties based on
the structure of the system constituents. The analysis has a local-to-global character.

Generalized cellular automata (GCA)

If, for example, the update scheme consists of applying the vertex functions synchronously one obtains the class of
generalized cellular automata (CA). In this case, the global map F: A — » a is given by

F{x) v = f v {x[v\) ■

This class is referred to as generalized cellular automata since the classical or standard cellular automata are typically

defined and studied over regular graphs or grids, and the vertex functions are typically assumed to be identical.

Example: Let Y be the circle graph on vertices {1,2,3,4} with edges {1,2}, {2,3}, {3,4} and {1,4}, denoted Circ .
Let K = {0,1} be the state space for each vertex and use the function nor : K — > K defined by
nor Jx,y,z) - (1 + x)(l +y)(l + z) with arithmetic modulo 2 for all vertex functions. Then for example the system
state (0,1,0,0) is mapped to (0, 0, 0, 1) using a synchronous update. All the transitions are shown in the phase space
below.

Graph dynamical system

216

0100 0001

1 100, 1010,
0110, 11/0,
1001,010), —
1101,0011,

1011,0111

1000 0010

0000

Mil

[2]

, w ) or

In this

Sequential dynamical systems (SDS)

If the vertex functions are applied asynchronously in the sequence specified by a word w = (w , w ,
permutation 7T = ( Tl, TTg, . . . , 7T„) of v[Y] one obtains the class of Sequential dynamical systems (SDS)
case it is convenient to introduce the 7-local maps F, constructed from the vertex functions by

Fi(x) = {x l3 X2, . . . ,Xi- lt fi(x[i]),x i+1 , . . . ,x n ) .

The SDS map F = [F , w] : K 11 — > K" is the function composition

[F Y ,w]

If the update sequence is a permutation one frequently speaks of a permutation SDS to emphasize this point.
Example: Let Y be the circle graph on vertices {1,2,3,4} with edges {1,2}, {2,3}, {3,4} and {1,4}, denoted Circ

F-m{m) ° F-w(m-l) ° '

F W (2) ° ^u(l) ■

Let ^={0,1} be the state space for each vertex and use the function nor : K

K defined by nor (x, y, z)

(1 + jc)(1 + y)(l + z) with arithmetic modulo 2 for all vertex functions. Using the update sequence (1,2,3,4) then the
system state (0, 1, 0, 0) is mapped to (0, 0, 1, 0). All the system state transitions for this sequential dynamical system
are shown in the phase space below.

(1234,1

-1000-

00 1 u /

0101

0000

10II-

nor

j 1 1 r

0010

OlOO-*-l00l

0001
0110

1110

Stochastic graph dynamical systems

From, e.g., the point of view of applications it is interesting to consider the case where one or more of the
components of a GDS contains stochastic elements. Motivating applications could include processes that are not
fully understood (e.g. dynamics within a cell) and where certain aspects for all practical purposes seem to behave
according to some probability distribution. There are also applications governed by deterministic principles whose
description is so complex or unwieldy that it makes sense to consider probabilistic approximations.

Every element of a graph dynamical system can be made stochastic in several ways. For example, in a sequential
dynamical system the update sequence can be made stochastic. At each iteration step one may choose the update
sequence w at random from a given distribution of update sequences with corresponding probabilities. The matching
probability space of update sequences induces a probability space of SDS maps. A natural object to study in this
regard is the Markov chain on state space induced by this collection of SDS maps. This case is referred to as update

Graph dynamical system 217

sequence stochastic GDS and is motivated by, e.g., processes where "events" occur at random according to certain
rates (e.g. chemical reactions), synchronization in parallel computation/discrete event simulations, and in
computational paradigms described later.

This specific example with stochastic update sequence illustrates two general facts for such systems: when passing to
a stochastic graph dynamical system one is generally led to (1) a study of Markov chains (with specific structure
governed by the constituents of the GDS), and (2) the resulting Markov chains tend to be large having an exponential
number of states. A central goal in the study of stochastic GDS is to be able to derive reduced models.

One may also consider the case where the vertex functions are stochastic, i.e., function stochastic GDS. For example,
Random Boolean networks are examples of function stochastic GDS using a synchronous update scheme and where
the state space is K = {0, 1 }. Finite probabilistic cellular automata (PC A) is another example of function stochastic
GDS. In principle the class of Interacting particle systems (IPS) covers finite and infinite PC A, but in practice the
work on IPS is largely concerned with the infinite case since this allows one to introduce more interesting topologies
on state space.

Applications

Graph dynamical systems constitute a natural framework for capturing distributed systems such as biological
networks and epidemics over social networks, many of which are frequently referred to as complex systems.

References

[1] Wu, Chai Wah (2005). "Synchronization in networks of nonlinear dynamical systems coupled via a directed graph". Nonlinearity 18 (3):

1057-1064. doi:10.1088/0951-7715/18/3/007.
[2] Mortveit, Henning S.; Reidys, Christian M. (2007). An introduction to sequential dynamical systems. Universitext. New York: Springer

Verlag. ISBN 978-0-387-30654-4.

External links

• Graph Dynamical Systems — A Mathematical Framework for Interaction-Based Systems, Their Analysis and
Simulations by Henning Mortveit (http://legacy.samsi.info/200809/algebraic/presentations/discrete/friday/
samsi-05-dec-08.pdf)

Further reading

• Macauley, Matthew; Mortveit, Henning S. (2009). "Cycle equivalence of graph dynamical systems". Nonlinearity
22 (2): 421-436. doi:10.1088/0951-7715/22/2/010.

• Golubitsky, Martin; Stewart, Ian (2003). The Symmetry Perspective. Basel: Birkhauser. ISBN 0817621717.

Analysis of Systems 218

Analysis of Systems

System analysis in the field of electrical engineering characterizes electrical systems and their properties. System
Analysis can be used to represent almost anything from population growth to audio speakers, electrical engineers
often use it because of its direct relevance to many areas of their discipline, most notably signal processing and
communication systems.

Characterization of systems

A system is characterized by how it responds to input signals. In general, a system has one or more input signals and
one or more output signals. Therefore, one natural characterization of systems is by how many inputs and outputs
they have:

• SISO (Single Input, Single Output)

• SIMO (Single Input, Multiple Outputs)

• MISO (Multiple Inputs, Single Output)

• MIMO (Multiple Inputs, Multiple Outputs)

It is often useful (or necessary) to break up a system into smaller pieces for analysis. Therefore, we can regard a
SIMO system as multiple SISO systems (one for each output), and similarly for a MIMO system. By far, the greatest
amount of work in system analysis has been with SISO systems, although many parts inside SISO systems have
multiple inputs (such as adders).

Signals can be continuous or discrete in time, as well as continuous or discrete in the values they take at any given
time:

• Signals that are continuous in time and continuous in value are known as analog signals.

• Signals that are discrete in time and discrete in value are known as digital signals.

• Signals that are discrete in time and continuous in value are called discrete-time signals. While important
mathematically, systems that process discrete time signals are difficult to physically realize. The methods
developed for analyzing discrete time signals and systems are usually applied to digital and analog signals and
systems.

• Signals that are continuous in time and discrete in value are sometimes seen in the timing analysis of logic
circuits, but have little to no use in system analysis.

With this categorization of signals, a system can then be characterized as to which type of signals it deals with:

• A system that has analog input and analog output is known as an analog system.

• A system that has digital input and digital output is known as a digital system.

• Systems with analog input and digital output or digital input and analog output are possible. However, it is usually
easiest to break these systems up for analysis into their analog and digital parts, as well as the necessary analog to
digital or digital to analog converter.

Another way to characterize systems is by whether their output at any given time depends only on the input at that
time or perhaps on the input at some time in the past (or in the future!).

• Memoryless systems do not depend on any past input.

• Systems with memory do depend on past input.

• Causal systems do not depend on any future input.

• Non-causal or anticipatory systems do depend on future input.

Note: It is not possible to physically realize a non-causal system operating in "real time". However, from the
standpoint of analysis, they are important for two reasons. First, the ideal system for a given application is
often a noncausal system, which although not physically possible can give insight into the design of a

Analysis of Systems 219

derivated causal system to accomplish a similar purpose. Second, there are instances when a system does not
operate in "real time" but is rather simulated "off-line" by a computer, such as post-processing an audio or
video recording.

Further, some non-causal systems can operate in pseudo-real time by introducing lag: if a system depends on
input for 1 second in future, it can process in real time with 1 second lag.

Analog systems with memory may be further classified as lumped or distributed. The difference can be explained by
considering the meaning of memory in a system. Future output of a system with memory depends on future input
and a number of state variables, such as values of the input or output at various times in the past. If the number of
state variables necessary to describe future output is finite, the system is lumped; if it is infinite, the system is
distributed.

Finally, systems may be characterized by certain properties which facilitate their analysis:

• A system is linear if it has the superposition and scaling properties.

• A system that is not linear is non-linear.

• If the output of a system does not depend explicitly on time, the system is said to be time-invariant; otherwise it is
time-variant

• A system that will always produce the same output for a given input is said to be deterministic.

• A system that will produce different outputs for a given input is said to be stochastic.

There are many methods of analysis developed specifically for linear time-invariant (LTI) deterministic systems.
Unfortunately, in the case of analog systems, none of these properties are ever perfectly achieved. Linearity implies
that operation of a system can be scaled to arbitrarily large magnitudes, which is not possible. Time-invariance is
violated by aging effects that can change the outputs of analog systems over time (usually years or even decades).
Thermal noise and other random phenomena ensure that the operation of any analog system will have some degree
of stochastic behavior. Despite these limitations, however, it is usually reasonable to assume that deviations from
these ideals will be small.

LTI Systems

As mentioned above, there are many methods of analysis developed specifically for LTI systems. This is due to their
simplicity of specification. An LTI system is completely specified by its transfer function (which is a rational
function for digital and lumped analog LTI systems). Alternatively, we can think of an LTI system being completely
specified by its frequency response. A third way to specify an LTI system is by its characteristic linear differential
equation (for analog systems) or linear difference equation (for digital systems). Which description is most useful
depends on the application.

The distinction between lumped and distributed LTI systems is important. A lumped LTI system is specified by a
finite number of parameters, be it the zeros and poles of its transfer function, or the coefficients of its differential
equation, whereas specification of a distributed LTI system requires a complete function

Dynamic Bayesian network 220

Dynamic Bayesian network

A dynamic Bayesian network is a Bayesian network that represents sequences of variables. These sequences are
often time-series (for example, in speech recognition) or sequences of symbols (for example, protein sequences). The
hidden Markov model and the Kalman filter can be considered as the most simple dynamic Bayesian networks.

References

• Learning Dynamic Bayesian Networks (1997), Zoubin Ghahramani, Lecture Notes In Computer Science, Vol.
1387, 168-197

References

[1] http://citeseerx.ist. psu.edu/viewdoc/download?doi=10. 1. 1.56.7874&rep=repl&type=pdf

Dynamic network analysis

Dynamic network analysis (DNA) is an emergent scientific field that brings together traditional social network
analysis (SNA), link analysis (LA) and multi-agent systems (MAS) within network science and network theory.
There are two aspects of this field. The first is the statistical analysis of DNA data. The second is the utilization of
simulation to address issues of network dynamics. DNA networks vary from traditional social networks in that they
are larger, dynamic, multi-mode, multi-plex networks, and may contain varying levels of uncertainty. The main
difference of DNA to SNA is DNA taken the domain of time into account. One of the most notable and earliest case
of the use of DNA is in Sampson's monastery study, where he took snapshots of the same network from different
intervals and observed and analyzed the evolution of the network.

DNA statistical tools are generally optimized for large-scale networks and admit the analysis of multiple networks
simultaneously in which, there are multiple types of nodes (multi-node) and multiple types of links (multi-plex). In
contrast, SNA statistical tools focus on single or at most two mode data and facilitate the analysis of only one type of
link at a time.

DNA statistical tools tend to provide more measures to the user, because they have measures that use data drawn
from multiple networks simultaneously. From a computer simulation perspective, nodes in DNA are like atoms in
quantum theory, nodes can be, though need not be, treated as probabilistic. Whereas nodes in a traditional SNA
model are static, nodes in a DNA model have the ability to learn. Properties change over time; nodes can adapt: A
company's employees can learn new skills and increase their value to the network; Or, capture one terrorist and three
more are forced to improvise. Change propagates from one node to the next and so on. DNA adds the element of a
network's evolution and considers the circumstances under which change is likely to occur.

Dynamic network analysis

221

Illustrative problems that « mB «, v
people in the DNA area
work on

• Developing metrics and statistics to
assess and identify change within
and across networks.

• Developing and validating
simulations to study network
change, evolution, adaptation,
decay. See Computer simulation
and organizational studies , , . ,. . ,. , ,

An example or a multi-entity, multi-network, dynamic network diagram

• Developing and testing theory of

network change, evolution, adaptation, decay

• Developing and validating formal models of network generation and evolution

• Developing techniques to visualize network change overall or at the node or group level

• Developing statistical techniques to see whether differences observed over time in networks are due to simply
different samples from a distribution of links and nodes or changes over time in the underlying distribution of
links and nodes

Developing control processes for networks over time

Developing algorithms to change distributions of links in networks over time

Developing algorithms to track groups in networks over time

Developing tools to extract or locate networks from various data sources such as texts

Developing statistically valid measurements on networks over time

Examining the robustness of network metrics under various types of missing data

Empirical studies of multi-mode multi-link multi-time period networks

Examining networks as probabilistic time-variant phenomena

Forecasting change in existing networks

Identifying trails through time given a sequence of networks

Identifying changes in node criticality given a sequence of networks anything else related to multi-mode
multi-link multi-time period networks

Kathleen Carley, of Carnegie Mellon University, is a leading authority in this field.

Further reading

• Kathleen M. Carley, 2003, "Dynamic Network Analysis" in Dynamic Social Network Modeling and Analysis:
Workshop Summary and Papers, Ronald Breiger, Kathleen Carley, and Philippa Pattison, (Eds.) Committee on
Human Factors, National Research Council, National Research Council. Pp. 133—145, Washington, DC.

• Kathleen M. Carley, 2002, "Smart Agents and Organizations of the Future" The Handbook of New Media. Edited
by Leah Lievrouw and Sonia Livingstone, Ch. 12, pp. 206—220, Thousand Oaks, CA, Sage.

• Kathleen M. Carley, Jana Diesner, Jeffrey Reminga, Maksim Tsvetovat, 2008, Toward an Interoperable Dynamic
Network Analysis Toolkit, DSS Special Issue on Cyberinfrastructure for Homeland Security: Advances in

121

Information Sharing, Data Mining, and Collaboration Systems. Decision Support Systems 43(4): 1324- 1347

Dynamic network analysis 222

(article 20 [3] )

• Terrill L. Frantz, Kathleen M. Carley. 2009, Toward A Confidence Estimate For The Most-Central-Actor
Finding. Academy of Management Annual Conference, Chicago, IL, USA, 7—11 August. (Awarded the Sage
Publications/RM Division Best Student Paper Award)

References

[1] Identity and control: a structural theory of social action By Harrison C. White

[2] http://www.sciencedirect.com/science/journal/01679236

[3] http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V8S-4KGG5P7-l&_user=4422&_coverDate=08%2F31%2F2007&
_rdoc=20&_fmt=high&_orig=browse&

_srch=doc-info(%23toc%235878%232007%23999569995%23665759%23FLA%23display%23Volume)&_cdi=5878&_sort=d&
_docanchor=&_ct=52&_acct=C000059600&_version=l&_urlVersion=0&_userid=4422&md5=9459e84d7a8863039c7abd5065266250

External links

• Radcliffe Exploratory Seminar on Dynamic Networks (http://www.eecs.harvard.edu/~parkes/
RadcliffeSeminar.htm)

• Center for Computational Analysis of Social and Organizational Systems (CASOS) (http://www.casos.cs.cmu.
edu/)

Dynamic circuit network

A dynamic circuit network (DCN) is an advanced computer networking technology that combines traditional
packet-switched communication based on the Internet Protocol, as used in the Internet, with circuit-switched
technologies that are characteristic of traditional telephone network systems. This combination allows user-initiated
ad-hoc dedicated allocation of network bandwidth for high-demand, real-time applications and network services,

delivered over an optical fiber infrastructure.

Implementation

T21
Dynamic circuit networks were pioneered by the Internet2 advanced networking consortium. The experimental

Internet2 HOPI infrastructure, decommissioned in 2007, was a forerunner to the current SONET-based Ciena

Network underlying the Internet2 DCN. The Internet2 DCN began operation in late 2007 as part of the larger

T31
Internet2 network. It provides advanced networking capabilities and resources to the scientific and research

communities, such as the Large Hadron Collider (LHC) project.

The Internet2 DCN is based on open-source, standards-based software, the Inter-domain Controller (IDC) protocol,
developed in cooperation with ESn
Network Software Suite (DCN SS).

developed in cooperation with ESnet and GEANT2. The entire software set is known as the Dynamic Circuit

Dynamic circuit network 223

Inter-domain Controller protocol

The Inter-domain Controller protocol manages the dynamic provisioning of network resources participating in a
dynamic circuit network across multiple administrative domain boundaries. It is a SOAP-based XML messaging
protocol, secured by Web Services Security (vl.l) using the XML Digital Signature standard. It is transported over
HTTP Secure (HTTPS) connections.

References

[1] Erica Naone (2008-02-14). "Bandwidth on Demand" (http://www.technologyreview.com/Infotech/20277/pagel/?a=f). MIT Technology

Review. .
[2] "Dynamic Circuit Network" (http://www.internet2.edu/network/dc/). Internet2. .
[3] Internet2 DCN Working Group (2009-02-03). "Internet2 DCN Pilot Service Definition" (https://spaces.internet2.edu/download/

attachments/ 1293 l/Internet2+DCN+Pilot+Service+Definition+v0.4.pdf) (PDF). .
[4] Mary E. Shacklett (2009-08-1 1). "Dynamic Circuit Network Debuts for Researchers" (http://www.internetevolution.com/author.

asp?section_id=562&doc_id=180322). Internet Evolution. . Retrieved 2009-08-19.
[5] C.P. Guok, D.W. Robinson, E. Chaniotakis, M.R. Thompson, W. Johnston, B. Tierney (2008). "A User Driven Dynamic Circuit Network

Implementation" (http://www.es.net/pub/esnet-doc/DANMS08_1569141354_Guok_et-al.pdf) (PDF). ESNET. .
[6] A. Lake, J. Vollbrecht, A. Brown, J. Zurawski, D. Robertson, M. Thompson, C. Guok, E. Chaniotakis, T. Lehman (2008-05-30).

"Inter-domain Controller (IDC) Protocol Specification" (https://wiki.internet2.edu/confluence/download/attachments/19074/

IDC-Messaging-draft.pdf?version=l) (PDF). .

External links

• Internet2 Website (http://www.internet2.edu/)

• Dynamic Circuit Network Suite (https://wiki.internet2.edu/confluence/display/DCNSS/Home)

Tensor product network

A tensor product network, in neural networks, is a network that exploits the properties of tensors to model
associative concepts such as variableassignment. Orthonormal vectors are chosen to model the ideas (such as
variable names and target assignments), and the tensor product of these vectors construct a network whose
mathematical properties allow the user to easily extract the association from it.

Cybernetics

224

Cybernetics

Cybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related to
control theory and systems theory. Both in its origins and in its evolution in the second half of the 20th century,
cybernetics is equally applicable to physical and social (that is, language-based) systems.

Overview

Cybernetics is most applicable when

the system being analysed is involved

in a closed signal loop; that is, where

action by the system causes some

change in its environment and that

change is fed to the system via

information (feedback) that causes the

system to adapt to these new

conditions: the system's changes affect

its behavior. This "circular causal"

relationship is necessary and sufficient

for a cybernetic perspective. System

Dynamics, a related field, originated

with applications of electrical

engineering control theory to other

kinds of simulation models (especially business systems) by Jay Forrester at MIT in the 1950s. Convenient GUI

system dynamics software developed into user friendly versions by the 1990s and have been applied to diverse

systems. SD models solve the problem of simultaneity (mutual causation) by updating all variables in small time

increments with positive and negative feedbacks and time delays structuring the interactions and control. The best

known SD model is probably the 1972 The Limits to Growth. This model forecast that exponential growth would

lead to economic collapse during the 21st century under a wide variety of growth scenarios.

Example of cybernetic thinking. On the one hand a company is approached as a system in
an environment. On the other hand cybernetic factory can be modeled as a control system.

Other fields of study which have influenced or been influenced by cybernetics include game theory, system theory (a
mathematical counterpart to cybernetics), perceptual control theory, sociology, psychology (especially
neuropsychology, behavioral psychology, cognitive psychology), philosophy, and architecture and organizational
theory.

Definition

„o N

v o0ioo io

°?T100^

The term cybernetics stems from the Greek KuPepvr|Tr|<; (kybernetes, steersman, governor,
pilot, or rudder — the same root as government). Cybernetics is a broad field of study, but
the essential goal of cybernetics is to understand and define the functions and processes of
systems that have goals and that participate in circular, causal chains that move from action
to sensing to comparison with desired goal, and again to action. Studies in cybernetics

Cybernetics

225

provide a means for examining the design and function of any system, including social systems such as business
management and organizational learning, including for the purpose of making them more efficient and effective.

Cybernetics was defined by Norbert Wiener, in his book of that title, as the study of control and communication in
the animal and the machine. Stafford Beer called it the science of effective organization and Gordon Pask extended it
to include information flows "in all media" from stars to brains. It includes the study of feedback, black boxes and
derived concepts such as communication and control in living organisms, machines and organizations including
self-organization. Its focus is how anything (digital, mechanical or biological) processes information, reacts to
information, and changes or can be changed to better accomplish the first two tasks .A more philosophical
definition, suggested in 1956 by Louis Couffignal, one of the pioneers of cybernetics, characterizes cybernetics as
"the art of ensuring the efficacy of action" . The most recent definition has been proposed by Louis Kauffman,
President of the American Society for Cybernetics, "Cybernetics is the study of systems and processes that interact

with themselves and produce themselves from themselves"

Concepts studied by cyberneticists (or, as some prefer, cyberneticians) include, but are not limited to: learning,
cognition, adaption, social control, emergence, communication, efficiency, efficacy and interconnect! vity. These
concepts are studied by other subjects such as engineering and biology, but in cybernetics these are removed from
the context of the individual organism or device.

Other fields of study which have influenced or been influenced by cybernetics include game theory; system theory (a
mathematical counterpart to cybernetics); psychology, especially neuropsychology, behavioral psychology and
cognitive psychology; philosophy; anthropology; and even theology, telematic art, and architecture.

History

The roots of cybernetic theory

The word cybernetics was first used in the context of "the study of self-governance" by Plato in The Laws to signify
the governance of people. The word 'cybernetique' was also used in 1834 by the physicist Andre-Marie Ampere
(1775—1836) to denote the sciences of government in his classification system of human knowledge.

The first artificial automatic regulatory system, a water clock, was invented
by the mechanician Ktesibios. In his water clocks, water flowed from a source
such as a holding tank into a reservoir, then from the reservoir to the
mechanisms of the clock. Ktesibios's device used a cone-shaped float to
monitor the level of the water in its reservoir and adjust the rate of flow of the
water accordingly to maintain a constant level of water in the reservoir, so
that it neither overflowed nor was allowed to run dry. This was the first
artificial truly automatic self-regulatory device that required no outside
intervention between the feedback and the controls of the mechanism.
Although they did not refer to this concept by the name of Cybernetics (they
considered it a field of engineering), Ktesibios and others such as Heron and
Su Song are considered to be some of the first to study cybernetic principles.

James Watt

The study of teleological mechanisms (from the Greek xzkoc, or telos for end,

goal, or purpose) in machines with corrective feedback dates from as far back

as the late 18th century when James Watt's steam engine was equipped with a

governor, a centrifugal feedback valve for controlling the speed of the engine. Alfred Russel Wallace identified this

as the principle of evolution in his famous 1858 paper. In 1868 James Clerk Maxwell published a theoretical article

on governors, one of the first to discuss and refine the principles of self-regulating devices. Jakob von Uexktill

Cybernetics 226

applied the feedback mechanism via his model of functional cycle {Funktionskreis) in order to explain animal
behaviour and the origins of meaning in general.

The early 20th century

Contemporary cybernetics began as an interdisciplinary study connecting the fields of control systems, electrical
network theory, mechanical engineering, logic modeling, evolutionary biology and neuroscience in the 1940s.
Electronic control systems originated with the 1927 work of Bell Telephone Laboratories engineer Harold S. Black
on using negative feedback to control amplifiers. The ideas are also related to the biological work of Ludwig von
Bertalanffy in General Systems Theory.

Early applications of negative feedback in electronic circuits included the control of gun mounts and radar antenna
during World War II. Jay Forrester, a graduate student at the Servomechanisms Laboratory at MIT during WWII
working with Gordon S. Brown to develop electronic control systems for the U.S. Navy, later applied these ideas to
social organizations such as corporations and cities as an original organizer of the MIT School of Industrial
Management at the MIT Sloan School of Management. Forrester is known as the founder of System Dynamics.

W. Edwards Deming, the Total Quality Management guru for whom Japan named its top post- WWII industrial prize,
was an intern at Bell Telephone Labs in 1927 and may have been influenced by network theory. Deming made
"Understanding Systems" one of the four pillars of what he described as "Profound Knowledge" in his book "The
New Economics."

Numerous papers spearheaded the coalescing of the field. In 1935 Russian physiologist P.K. Anokhin published a
book in which the concept of feedback ("back afferentation") was studied. The study and mathematical modelling of
regulatory processes became a continuing research effort and two key articles were published in 1943. These papers
were "Behavior, Purpose and Teleology" by Arturo Rosenblueth, Norbert Wiener, and Julian Bigelow; and the paper
"A Logical Calculus of the Ideas Immanent in Nervous Activity" by Warren McCulloch and Walter Pitts.

Cybernetics as a discipline was firmly established by Wiener, McCulloch and others, such as W. Ross Ashby and W.
Grey Walter.

Walter was one of the first to build autonomous robots as an aid to the study of animal behaviour. Together with the
US and UK, an important geographical locus of early cybernetics was France.

In the spring of 1947, Wiener was invited to a congress on harmonic analysis, held in Nancy, France. The event was
organized by the Bourbaki, a French scientific society, and mathematician Szolem Mandelbrojt (1899—1983), uncle
of the world-famous mathematician Benoit Mandelbrot.

Cybernetics

227

During this stay in France, Wiener received the offer to write a
manuscript on the unifying character of this part of applied
mathematics, which is found in the study of Brownian motion and in
telecommunication engineering. The following summer, back in the
United States, Wiener decided to introduce the neologism cybernetics
into his scientific theory. The name cybernetics was coined to denote
the study of "teleological mechanisms" and was popularized through
his book Cybernetics, or Control and Communication in the Animal
and Machine (Hermann & Cie, Paris, 1948). In the UK this became the
focus for the Ratio Club.

In the early 1940s John von Neumann, although better known for his

work in mathematics and computer science, did contribute a unique

and unusual addition to the world of cybernetics: Von Neumann

cellular automata, and their logical follow up the Von Neumann

Universal Constructor. The result of these deceptively simple

thought-experiments was the concept of self replication which

cybernetics adopted as a core concept. The concept that the same properties of genetic reproduction applied to social

memes, living cells, and even computer viruses is further proof of the somewhat surprising universality of cybernetic

study.

Wiener popularized the social implications of cybernetics, drawing analogies between automatic systems (such as a
regulated steam engine) and human institutions in his best-selling The Human Use of Human Beings : Cybernetics
and Society (Houghton-Mifflin, 1950).

While not the only instance of a research organization focused on cybernetics, the Biological Computer Lab at the
University of Illinois, Urbana/Champaign, under the direction of Heinz von Foerster, was a major center of

ro]

cybernetic research for almost 20 years, beginning in 1958.

John von Neumann

The fall and rebirth of cybernetics

For a time during the past 30 years, the field of cybernetics followed a boom-bust cycle of becoming more and more
dominated by the subfields of artificial intelligence and machine-biological interfaces (i.e. cyborgs) and when this
research fell out of favor, the field as a whole fell from grace.

In the 1970s new cyberneticians emerged in multiple fields, but especially in biology. The ideas of Maturana, Varela
and Atlan, according to Dupuy (1986) "realized that the cybernetic metaphors of the program upon which molecular
biology had been based rendered a conception of the autonomy of the living being impossible. Consequently, these
thinkers were led to invent a new cybernetics, one more suited to the organizations which mankind discovers in
nature - organizations he has not himself invented" . However, during the 1980s the question of whether the
features of this new cybernetics could be applied to social forms of organization remained open to debate.

In political science, Project Cybersyn attempted to introduce a cybernetically controlled economy during the early
1970s. In the 1980s, according to Harries-Jones (1988) "unlike its predecessor, the new cybernetics concerns itself
with the interaction of autonomous political actors and subgroups, and the practical and reflexive consciousness of
the subjects who produce and reproduce the structure of a political community. A dominant consideration is that of
recursiveness, or self -reference of political action both with regards to the expression of political consciousness and
with the ways in which systems build upon themselves".

One characteristic of the emerging new cybernetics considered in that time by Geyer and van der Zouwen, according
to Bailey (1994), was "that it views information as constructed and reconstructed by an individual interacting with
the environment. This provides an epistemological foundation of science, by viewing it as observer-dependent.

Cybernetics

228

Another characteristic of the new cybernetics is its contribution towards bridging the "micro-macro gap". That is, it
links the individual with the society" Another characteristic noted was the "transition from classical cybernetics to
the new cybernetics [that] involves a transition from classical problems to new problems. These shifts in thinking
involve, among others, (a) a change from emphasis on the system being steered to the system doing the steering, and
the factor which guides the steering decisions.; and (b) new emphasis on communication between several systems
which are trying to steer each other" . The work of Gregory Bateson was also strongly influenced by cybernetics.

Recent endeavors into the true focus of cybernetics, systems of control and emergent behavior, by such related fields
as game theory (the analysis of group interaction), systems of feedback in evolution, and metamaterials (the study of
materials with properties beyond the Newtonian properties of their constituent atoms), have led to a revived interest

in this increasingly relevant field

[2]

Subdivisions of the field

Cybernetics is an earlier but still-used generic term for many types of subject matter. These subjects also extend into
many others areas of science, but are united in their study of control of systems.

Basic cybernetics

Cybernetics studies systems of control as a concept, attempting to discover the basic principles underlying such
things as

Artificial intelligence
Robotics
Computer Vision
Control systems
Emergence
Learning organization
New Cybernetics
Second-order cybernetics
Interactions of Actors Theory

^_ . _. ASIMO uses sensors and intelligent algorithms to

Conversation Theory

avoid obstacles and navigate stairs.

In biology

Cybernetics in biology is the study of cybernetic systems present in biological organisms, primarily focusing on how
animals adapt to their environment, and how information in the form of genes is passed from generation to

1121

generation . There is also a secondary focus on combining artificial systems with biological systems.

Bioengineering
Biocybemetics
Bionics
Homeostasis
Medical cybernetics
Synthetic Biology
Systems Biology

Cybernetics

229

In computer science

Computer science directly applies the concepts of cybernetics to the control of devices and the analysis of
information.

• Robotics

• Decision support system

• Cellular automaton

• Simulation

• Technology

In engineering

Cybernetics in engineering is used to analyze cascading failures and System Accidents, in which the small errors and
imperfections in a system can generate disasters. Other topics studied include:

• Adaptive systems

• Engineering cybernetics

• Ergonomics

• Biomedical engineering

• Systems engineering

In management

Entrepreneurial cybernetics
Management cybernetics
Organizational cybernetics

Operations research
Systems engineering

An artificial heart, a product of biomedical
engineering.

In mathematics

Mathematical Cybernetics focuses on the factors of information, interaction of parts in systems, and the structure of
systems.

• Dynamical system

• Information theory

• Systems theory

In psychology

• Homunculus

• Psycho-Cybernetics

• Systems psychology

• Perceptual Control Theory

In sociology

By examining group behavior through the lens of cybernetics, sociology seeks the reasons for such spontaneous
events as smart mobs and riots, as well as how communities develop rules, such as etiquette, by consensus without
formal discussion. Affect Control Theory explains role behavior, emotions, and labeling theory in terms of
homeostatic maintenance of sentiments associated with cultural categories. The most comprehensive attempt ever
made in the social sciences to increase cybernetics in a generalized theory of society was made by Talcott Parsons.

Cybernetics 230

In this way, cybernetics establish the basic hierarchy in Parsons' AGIL paradigm, which is the ording
system-dimension of his action theory. These and other cybernetic models in sociology are reviewed in a book edited

ri3i

by McClelland and Fararo

• Affect Control Theory

• Memetics

• Sociocybernetics

In art

The artist Roy Ascott theorised the cybernetics of art in "Behaviourist Art and the Cybernetic Vision". Cybernetica,
Journal of the International Association for Cybernetics (Namur), 1967.

• Telematic art

• Interactive Art

• Systems art

Related fields
Complexity science

Complexity science attempts to understand the nature of complex systems.

• Complex Adaptive System

• Complex systems

• Complexity theory

References

[I] Tange, Kenzo (1966) "Function, Structure and Symbol".

[2] Kelly, Kevin (1994). Out of control: The new biology of machines, social systems and the economic world. Boston: Addison- Wesley.

ISBN 0-201-48340-8. OCLC 221860672 32208523 40868076 56082721 57396750.
[3] Couffignal, Louis, "Essai d'une definition generale de la cybernetique", The First International Congress on Cybernetics, Namur, Belgium,

June 26-29, 1956, Gauthier-Villars, Paris, 1958, pp. 46-54
[4] CYBCON discusstion group 20 September 2007 18: 15

[5] Granfield, Patrick (1973). Ecclesial Cybernetics: A Study of Democracy in the Church. New York: MacMillan. pp. 280.
[6] Hight, Christopher (2007). Architectural Principles in the age of Cybernetics. Routledge. pp. 248. ISBN 978-0415384827.
[7] http://www.ece.uiuc.edu/pubs/bcl/mueller/index.htm
[8] http://www.ece.uiuc.edu/pubs/bcl/hutchinson/index.htm
[9] Jean-Pierre Dupuy, "The autonomy of social reality: on the contribution of systems theory to the theory of society" in: Elias L. Khalil &

Kenneth E. Boulding eds., Evolution, Order and Complexity, 1986.
[10] Peter Harries-Jones (1988), "The Self-Organizing Polity: An Epistemological Analysis of Political Life by Laurent Dobuzinskis" in:

Canadian Journal of Political Science (Revue canadienne de science politique), Vol. 21, No. 2 (Jun., 1988), pp. 431-433.

[II] Kenneth D. Bailey (1994), Sociology and the New Systems Theory: Toward a Theoretical Synthesis, p. 163.

[12] Note: this does not refer to the concept of Racial Memory but to the concept of cumulative adaptation to a particular niche, such as the case

of the pepper moth having genes for both light and dark environments.
[13] McClelland, Kent A., and Thomas J. Fararo (Eds.). 2006. Purpose, Meaning, and Action: Control Systems Theories in Sociology. New

York: Palgrave Macmillan.

Cybernetics 23 1

Further reading

Andrew Pickering (2010) The Cybernetic Brain: Sketches of Another Future (http://www.amazon.com/

Cybernetic-Brain-Sketches-Another-Future/dp/0226667898) University Of Chicago Press.

Slava Gerovitch (2002) From Newspeak to Cyberspeak: A History of Soviet Cybernetics (http://web.mit.edu/

slava/homepage/newspeak.htm) MIT Press.

John Johnston, (2008) "The Allure of Machinic Life: Cybernetics, Artificial Life, and the New AI", MIT Press

Heikki Hyotyniemi (2006). Neocybernetics in Biological Systems (http://neocybernetics.com/reportl51/).

Espoo: Helsinki University of Technology, Control Engineering Laboratory.

Eden Medina, "Designing Freedom, Regulating a Nation: Socialist Cybernetics in Allende's Chile." Journal of

Latin American Studies 38 (2006):57 1-606.

Lars Bluma, (2005), Norbert Wiener und die Entstehung der Kybernetik im Zweiten Weltkrieg, Mlinster.

Francis Heylighen, and Cliff Joslyn (2001). " Cybernetics and Second Order Cybernetics (http://pespmcl.vub.

ac.be/Papers/Cybernetics-EPST.pdf)", in: R.A. Meyers (ed.), Encyclopedia of Physical Science & Technology

(3rd ed.), Vol. 4, (Academic Press, New York), p. 155-170.

Charles Francois (1999). " Systemics and cybernetics in a historical perspective (http://www.uni-klu.ac.at/

~gossimit/ifsr/francois/papers/systemics_and_cybernetics_in_a_historical_perspective.pdf)". In: Systems

Research and Behavioral Science. Vol 16, pp. 203—219 (1999)

Heinz von Foerster, (1995), Ethics and Second-Order Cybernetics (http://www.stanford.edu/group/SHR/4-2/

text/foerster. html) .

Steve J. Heims (1993), Constructing a Social Science for Postwar America. The Cybernetics Group, 1946-1953,

Cambridge University Press, London, UK.

Paul Pangaro (1990), "Cybernetics — A Definition", Eprint (http://pangaro.com/published/cyber-macmillan.

html).

Stuart Umpleby (1989), "The science of cybernetics and the cybernetics of science" (ftp://ftp.vub.ac.be/pub/

projects/Principia_Cybernetica/Papers_Umpleby/Science-Cybernetics.txt), in: Cybernetics and Systems", Vol.

21, No. 1, (1990), pp. 109-121.

Michael A. Arbib (1987, 1964) Brains, Machines, and Mathematics (http://www.amazon.com/

Brains-Machines-Mathematics-Michael-Arbib/dp/0387965394) Springer.

B.C. Patten, and E.P. Odum(1981), "The Cybernetic Nature of Ecosystems", The American Naturalist 118,

886-895.

Hans Joachim Ilgauds (1980), Norbert Wiener, Leipzig.

Steve J. Heims (1980), John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life

and Death, 3. Aufl., Cambridge.

Stafford Beer (1974), Designing Freedom, John Wiley, London and New York, 1975.

Gordon Pask (1972), " Cybernetics (http://www.cybsoc.org/gcyb.htm)", entry in Encyclopaedia Britannica

1972.

Helvey, T.C. The Age of Information: An Interdisciplinary Survey of Cybernetics. Englewood Cliffs, N.J.:

Educational Technology Publications, 1971.

Roy Ascott (1967). Behaviourist Art and the Cybernetic Vision. Cybernetica, Journal of the International

Association for Cybernetics (Namur), 10, pp. 25—56

W. Ross Ashby (1956), Introduction to Cybernetics. Methuen, London, UK. PDF text (http://pespmcl.vub.ac.

be/books/IntroCyb . pdf) .

Norbert Wiener (1948), Cybernetics or Control and Communication in the Animal and the Machine, (Hermann &

Cie Editeurs, Paris, The Technology Press, Cambridge, Mass., John Wiley & Sons Inc., New York, 1948).

Cybernetics

232

External links

General

• Norbert Wiener and Stefan Odobleja - A Comparative Analysis (http://www.bu.edu/wcp/Papers/Comp/
CompJurc.htm)

• Reading List for Cybernetics (http://www.cscs.umich.edu/~crshalizi/notabene/cybernetics.html)

• Principia Cybernetica Web (http://pespmcl.vub.ac.be/DEFAULT.html)

• Web Dictionary of Cybernetics and Systems (http://pespmcl.vub.ac.be/ASC/indexASC.html)

• Glossary Slideshow (136 slides) (http://www.gwu.edu/~asc/slide/sl.html)

• Basics of Cybernetics (http://www.smithsrisca.demon.co.uk/cybernetics.html)

• What is Cybernetics? (http://www.youtube.com/watch?v=_hjAXkNbPfk) Livas short introductory videos on
YouTube

• A History of Systemic and Cybernetic Thought. From Homeostasis to the Teardrop (http://www.pclibya.com/
cybernetic_teardrop.pdf)

Societies

• American Society for Cybernetics (http://www.asc-cybernetics.org/)

• IEEE Systems, Man, & Cybernetics Society (http://www.ieeesmc.org/)

• The Cybernetics Society (http://www.cybsoc.org)

Systems Biology

Systems biology is a term used to

describe a number of trends in

bioscience research, and a movement

which draws on those trends.

Proponents describe systems biology as

a biology-based inter-disciplinary study

field that focuses on complex

interactions in biological systems,

claiming that it uses a new perspective

(holism instead of reduction).

Particularly from year 2000 onwards,

the term is used widely in the

biosciences, and in a variety of

contexts. An often stated ambition of

systems biology is the modeling and

discovery of emergent properties, properties of a system whose theoretical description is only possible using

techniques which fall under the remit of systems biology.

Systems Biology 233

Overview

Systems biology can be considered from a number of different aspects:

• As a field of study, particularly, the study of the interactions between the components of biological systems, and
how these interactions give rise to the function and behavior of that system (for example, the enzymes and
metabolites in a metabolic pathway).

• As a paradigm, usually defined in antithesis to the so-called reductionist paradigm (biological organisation),
although fully consistent with the scientific method. The distinction between the two paradigms is referred to in
these quotations:

"The reductionist approach has successfully identified most of the components and many of the interactions

but, unfortunately, offers no convincing concepts or methods to understand how system properties emerge. ..the

pluralism of causes and effects in biological networks is better addressed by observing, through quantitative

measures, multiple components simultaneously and by rigorous data integration with mathematical models"
c • [3]

Science

"Systems biology. ..is about putting together rather than taking apart, integration rather than reduction. It
requires that we develop ways of thinking about integration that are as rigorous as our reductionist
programmes, but different. ...It means changing our philosophy, in the full sense of the term" Denis Noble

• As a series of operational protocols used for performing research, namely a cycle composed of theory,
analytic or computational modelling to propose specific testable hypotheses about a biological system,
experimental validation, and then using the newly acquired quantitative description of cells or cell processes to
refine the computational model or theory. Since the objective is a model of the interactions in a system, the
experimental techniques that most suit systems biology are those that are system-wide and attempt to be as
complete as possible. Therefore, transcriptomics, metabolomics, proteomics and high-throughput techniques are
used to collect quantitative data for the construction and validation of models.

• As the application of dynamical systems theory to molecular biology.

• As a socioscientific phenomenon defined by the strategy of pursuing integration of complex data about the
interactions in biological systems from diverse experimental sources using interdisciplinary tools and personnel.

This variety of viewpoints is illustrative of the fact that systems biology refers to a cluster of peripherally
overlapping concepts rather than a single well-delineated field. However the term has widespread currency and
popularity as of 2007, with chairs and institutes of systems biology proliferating worldwide.

History

Systems biology finds its roots in:

• the quantitative modeling of enzyme kinetics, a discipline that flourished between 1900 and 1970,

• the mathematical modeling of population growth,

• the simulations developed to study neurophysiology, and

• control theory and cybernetics.

One of the theorists who can be seen as one of the precursors of systems biology is Ludwig von Bertalanffy with his
general systems theory. One of the first numerical simulations in biology was published in 1952 by the British
neurophysiologists and Nobel prize winners Alan Lloyd Hodgkin and Andrew Fielding Huxley, who constructed a
mathematical model that explained the action potential propagating along the axon of a neuronal cell. Their model
described a cellular function emerging from the interaction between two different molecular components, a
potassium and a sodium channels, and can therefore be seen as the beginning of computational systems biology. In
1960, Denis Noble developed the first computer model of the heart pacemaker.

Systems Biology

234

The formal study of systems biology, as a distinct discipline, was launched by systems theorist Mihajlo Mesarovic in
1966 with an international symposium at the Case Institute of Technology in Cleveland, Ohio entitled "Systems
Theory and Biology."

The 1960s and 1970s saw the development of several approaches to study complex molecular systems, such as the
Metabolic Control Analysis and the biochemical systems theory. The successes of molecular biology throughout the
1980s, coupled with a skepticism toward theoretical biology, that then promised more than it achieved, caused the
quantitative modelling of biological processes to become a somewhat minor field.

However the birth of functional genomics in the 1990s meant that large quantities of high quality data became
available, while the computing power exploded, making more realistic models possible. In 1997, the group of
Masaru Tomita published the first quantitative model of the metabolism of a whole (hypothetical) cell.

Around the year 2000, after Institutes of Systems Biology were established in Seattle and Tokyo, systems biology
emerged as a movement in its own right, spurred on by the completion of various genome projects, the large increase
in data from the omics (e.g. genomics and proteomics) and the accompanying advances in high-throughput
experiments and bioinformatics. Since then, various research institutes dedicated to systems biology have been

ri3i

developed. As of summer 2006, due to a shortage of people in systems biology several doctoral training centres in
systems biology have been established in many parts of the world.

Disciplines associated with systems biology

According to the interpretation of Systems
Biology as the ability to obtain, integrate and
analyze complex data from multiple
experimental sources using interdisciplinary
tools, some typical technology platforms are:

• Phenomics: Organismal variation in
phenotype as it changes during its life span.

• Genomics: Organismal deoxyribonucleic acid
(DNA) sequence, including intra-organisamal
cell specific variation, (i.e. Telomere length
variation etc.).

Overview of signal transduction pathways

• Epigenomics / Epigenetics: Organismal and
corresponding cell specific transcriptomic
regulating factors not empirically coded in the
genomic sequence, (i.e. DNA methylation, Histone Acetelation etc.).

• Transcriptomics: Organismal, tissue or whole cell gene expression measurements by DNA microarrays or serial
analysis of gene expression

• Interferomics: Organismal, tissue, or cell level transcript correcting factors (i.e. RNA interference)

• Translatomics / Proteomics: Organismal, tissue, or cell level measurements of proteins and peptides via
two-dimensional gel electrophoresis, mass spectrometry or multi-dimensional protein identification techniques
(advanced HPLC systems coupled with mass spectrometry). Sub disciplines include phosphoproteomics,
glycoproteomics and other methods to detect chemically modified proteins.

• Metabolomics: Organismal, tissue, or cell level measurements of all small-molecules known as metabolites.

• Glycomics: Organismal, tissue, or cell level measurements of carbohydrates.

• Lipidomics: Organismal, tissue, or cell level measurements of lipids.

In addition to the identification and quantification of the above given molecules further techniques analyze the
dynamics and interactions within a cell. This includes:

Systems Biology 235

• Interactomics: Organismal, tissue, or cell level study of interactions between molecules. Currently the
authoritative molecular discipline in this field of study is protein-protein interactions (PPI), although the working
definition does not pre-clude inclusion of other molecular disciplines such as those defined here.

• Fluxomics: Organismal, tissue, or cell level measurements of molecular dynamic changes over time.

• Biomics: systems analysis of the biome.

The investigations are frequently combined with large scale perturbation methods, including gene-based (RNAi,
mis-expression of wild type and mutant genes) and chemical approaches using small molecule libraries. Robots and
automated sensors enable such large-scale experimentation and data acquisition. These technologies are still
emerging and many face problems that the larger the quantity of data produced, the lower the quality. A wide variety
of quantitative scientists (computational biologists, statisticians, mathematicians, computer scientists, engineers, and
physicists) are working to improve the quality of these approaches and to create, refine, and retest the models to
accurately reflect observations.

The systems biology approach often involves the development of mechanistic models, such as the reconstruction of
dynamic systems from the quantitative properties of their elementary building blocks. For instance, a cellular

network can be modelled mathematically using methods coming from chemical kinetics and control theory. Due to
the large number of parameters, variables and constraints in cellular networks, numerical and computational
techniques are often used.

Other aspects of computer science and informatics are also used in systems biology. These include:

• New forms of computational model, such as the use of process calculi to model biological processes (notable
approaches include stochastic 7T -calculus, BioAmbients, Beta Binders, BioPEPA and Brane calculus) and
constraint-based modeling.

• Integration of information from the literature, using techniques of information extraction and text mining.

• Development of online databases and repositories for sharing data and models, approaches to database integration
and software interoperability via loose coupling of software, websites and databases, or commercial suits.

• Development of syntactically and semantically sound ways of representing biological models.

References

[I] Snoep J.L. and Westerhoff H.V.; Alherghina L. and Westerhoff H.V. (Eds.) (2005). "From isolation to integration, a systems biology
approach for building the Silicon Cell". Systems Biology: Definitions and Perspectives. Springer- Verlag. p. 7.

[2] "Systems Biology — the 21st Century Science" (http://www.systemsbiology.org/Intro_to_ISB_and_Systems_Biology/

Systems_Biology_— _the_21st_Century_Science). .
[3] Sauer, U. et al. (27 April 2007). "Getting Closer to the Whole Picture". Science 316: 550. doi: 10.1 126/science. 1 142502. PMTD 17463274.
[4] Denis Noble (2006). The Music of Life: Biology beyond the genome. Oxford University Press. ISBN 978-0199295739. p21
[5] "Systems Biology: Modelling, Simulation and Experimental Validation" (http://www.bbsrc.ac.uk/science/areas/ebs/themes/

main_sysbio.html). .
[6] Kholodenko B.N., Bruggeman F.J., Sauro H.M.; Alberghina L. and Westerhoff H.V. (Eds.) (2005). "Mechanistic and modular approaches to

modeling and inference of cellular regulatory networks". Systems Biology: Definitions and Perspectives. Springer- Verlag. p. 143.
[7] von Bertalanffy, Ludwig (1968). General System theory: Foundations, Development, Applications. George Braziller. ISBN 0807604534.
[8] Hodgkin AL, Huxley AF (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". J

Physiol 117 (4): 500-544. PMC 1392413. PMID 12991237.
[9] Le Novere, N (2007). "The long journey to a Systems Biology of neuronal function". BMC Systems Biology 1: 28.

doi: 10.1 186/1752-0509-1-28. PMC 1904462. PMID 17567903.
[10] Noble D (1960). "Cardiac action and pacemaker potentials based on the Hodgkin-Huxley equations". Nature 188 (4749): 495—497.

doi:10.1038/188495b0. PMID 13729365.

[II] Mesarovic, M. D. (1968). Systems Theory and Biology. Springer- Verlag.

[12] "A Means Toward a New Holism" (http://www.jstor.Org/view/00368075/ap004022/00a00220/0). Science 161 (3836): 34-35.

doi: 10.1 126/science.l61.3836.34. .
[13] "Working the Systems" (http://sciencecareers.sciencemag.org/career_development/previous_issues/articles/2006_03_03/

working_the_systems/ (parent)/ 158). .
[14] Gardner, TS; di Bernardo D, Lorenz D and Collins JJ (4 July 2003). "Inferring genetic networks and identifying compound of action via

expression profiling". Science 301: 102-1005. doi: 10.1 126/science. 108 1900. PMID 12843395.

Systems Biology 236

[15] di Bernardo, D; Thompson MJ, Gardner TS, Chobot SE, Eastwood EL, Wojtovich AP, Elliot SJ, Schaus SE and Collins JJ (March 2005).
"Chemogenomic profiling on a genome-wide scale using reverse-engineered gene networks". Nature Biotechnology 23: 377—383.
doi:10.1038/nbtl075. PMID 15765094.

Further reading

• Bear, M. F.; B. W. Connors, and M. A. Paradiso (2006). Neuroscience: Exploring the Brain (3rd ed.).
Philadelphia: Lippincott. ISBN 0781760038.

• Binder/Hirokaw a/Windhorst (2009, 4399pp, 5 vols). Encyclopedia of Neuroscience (http://www.springer.com/
biomed/neuroscience/book/978-3-540-23735-8). Springer. ISBN 978-3-540-23735-8.

• Kandel, ER; Schwartz JH, Jessell TM (2000). Principles of Neural Science (4th ed.). New York: McGraw-Hill.
ISBN 0-8385-7701-6.

• Squire, L. et al. (2003). Fundamental Neuroscience, 2nd edition. Academic Press; ISBN 0-12-660303-0

• Byrne and Roberts (2004). From Molecules to Networks. Academic Press; ISBN 0-12-148660-5

• Sanes, Reh, Harris (2005). Development of the Nervous System, 2nd edition. Academic Press; ISBN
0-12-618621-9

• Siegel et al. (2005). Basic Neurochemistry, 7th edition. Academic Press; ISBN 0-12-088397-X

• Rieke, F. et al. (1999). Spikes: Exploring the Neural Code. The MIT Press; Reprint edition ISBN 0-262-68108-0

• section.47 Neuroscience (http://www.ncbi.nlm. nih.gov/entrez/query. fcgi?cmd=Search&db=books&
doptcmdl=GenBookHL&term=The+Cellular+Components+of+the+Nervous+System+AND+
neurosci[book]+AND+231002[uid]&rid=neurosci.) 2nd ed. Dale Purves, George J. Augustine, David
Fitzpatrick, Lawrence C. Katz, Anthony-Samuel LaMantia, James O. McNamara, S. Mark Williams. Published
by Sinauer Associates, Inc., 2001.

• section. 18 Basic Neurochemistry: Molecular, Cellular, and Medical Aspects (http://www.ncbi.nlm.nih.gov/
entrez/query.fcgi?cmd=Search&db=books&doptcmdl=GenBookHL&term=Characteristics+of+the+Neuron+
AND+bnchm[book]+AND+160014[uid]&rid=bnchm.) 6th ed. by George J. Siegel, Bernard W. Agranoff, R.
Wayne Albers, Stephen K. Fisher, Michael D. Uhler, editors. Published by Lippincott, Williams & Wilkins, 1999.

• Andreasen, Nancy C. (March 4 2004). Brave New Brain: Conquering Mental Illness in the Era of the Genome
(http://www.oup.com/uk/catalogue/?ci=9780195 145090). Oxford University Press. ISBN 9780195145090.

• Damasio, A. R. (1994). Descartes' Error: Emotion, Reason, and the Human Brain. New York, Avon Books.
ISBN 0-399-13894-3 (Hardcover) ISBN 0-380-72647-5 (Paperback)

• Gardner, H. (1976). The Shattered Mind: The Person After Brain Damage. New York, Vintage Books, 1976
ISBN 0-394-71946-8

• Goldstein, K. (2000). The Organism. New York, Zone Books. ISBN 0-942299-96-5 (Hardcover) ISBN
0-942299-97-3 (Paperback)

• Lauwereyns, Jan (February 2010). The Anatomy of Bias: How Neural Circuits Weigh the Options (http://
mitpress.mit.edu/9780262123105). Cambridge, MA: The MIT Press. ISBN 026212310X.

• Llinas R. (2001). / of the Vortex: From Neurons to Self MIT Press. ISBN 0-262-12233-2 (Hardcover) ISBN
0-262-62163-0 (Paperback)

Neurosciences 246

Luria, A. R. (1997). The Man with a Shattered World: The History of a Brain Wound. Cambridge, Massachusetts,

Harvard University Press. ISBN 0-224-00792-0 (Hardcover) ISBN 0-674-54625-3 (Paperback)

Luria, A. R. (1998). The Mind of a Mnemonist: A Little Book About A Vast Memory. New York, Basic Books, Inc.

ISBN 0-674-57622-5

Medina, J. (2008). Brain Rules: 12 Principles for Surviving and Thriving at Work, Home, and School. Seattle,

Pear Press. ISBN 0-979-777704 (Hardcover with DVD)

Pinker, S. (1999). How the Mind Works. W. W. Norton & Company. ISBN 0-393-31848-6

Pinker, S. (2002). The Blank Slate: The Modern Denial of Human Nature. Viking Adult. ISBN 0-670-03151-8

Robinson, D. L. (2009). Brain, Mind and Behaviour: A New Perspective on Human Nature (2nd ed.). Dundalk,

Ireland: Pontoon Publications. ISBN 978-0-9561812-0-6.

Ramachandran, V. S. (1998). Phantoms in the Brain. New York, New York Harper Collins. ISBN 0-688-15247-3

(Paperback)

Rose, S. (2006). 21st Century Brain: Explaining, Mending & Manipulating the Mind ISBN 0099429772

(Paperback)

Sacks, O. The Man Who MistookHis Wife for a Hat. Summit Books ISBN 0-671-55471-9 (Hardcover) ISBN

0-06-097079-0 (Paperback)

Sacks, O. (1990). Awakenings. New York, Vintage Books. (See also Oliver Sacks) ISBN 0-671-64834-9

(Hardcover) ISBN 0-06-097368-4 (Paperback)

Sternberg, E. (2007) Are You a Machine? The Brain, the Mind and What it Means to be Human. Amherst, NY:

Prometheus Books.

External links

• Neuroscience (http://www.bbc.co.uk/programmes/b00fbd26) on In Our Time at the BBC. ( listen now (http:/
/www. bbc.co.uk/iplayer/console/b00fbd26/In_Our_Time_Neuroscience))
Neuroscience Information Framework (NIF) (http://www.neuinfo.org)

Neurobiology (http://www.dmoz.org/Science/Biology/Neurobiology/) at the Open Directory Project
IBRO (International Brain Research Organization) (http://www.ibro.info)
Society for Neuroscience (SFN) (http://www.sfn.org)
American Society for Neurochemistry (http://www.asneurochem.org)
Neuroscience Online (electronic neuroscience textbook) (http://neuroscience.uth.tmc.edu/)
Faculty for Undergraduate Neuroscience (FUN) (http://www.funfaculty.org/drupal/)
Neuroscience for Kids (http://faculty.washington.edu/chudler/neurok.html)

Neuroscience Discussion Group (https://www.researchgate.net/group/Neuroscience) in ResearchGate
Neuroscience Discussion Forum (http://www.neuroscienceforums.com)

HHMI Neuroscience lecture series - Making Your Mind: Molecules, Motion, and Memory (http://www.hhmi.
org/biointeractive/neuroscience/lectures.html)
British Neuroscience Association (http://www.bna.org.uk)

Biocybernetics 247

Biocybernetics

Biocybernetics is the application of cybernetics to biological science, composed of biological disciplines that benefit
from the application of cybernetics: neurology, multicellular systems and others. Biocybernetics plays a major role in
systems biology, seeking to integrate different levels of information to understand how biological systems function.

Biocybernetics as an abstract science is a part of theoretical biology, and based upon the principles of systemics.

Terminology

Biocybernetics is a cojoined word from bio (Greek: (3(,o / life) and cybernetics (Greek: Kupspv^TLKT) /
controlling-governing). It is sometimes written together or with a blank or written fully as biological cybernetics,
whilst the same rules apply. Most write it together though, as Google statistics show. The same applies to neuro
cybernetics which should also be looked up as neurological, when doing extensive research.

Early fathers of biocybernetics

Ross Ashby, 1956 [1]
Hans Drischel, 1972 [2]
Norbert Wiener, 1948 [3]

Same or familiar fields

As those disciplines are dealing on theoretical/abstract foundations and are in accordance with the popularity of
computers. Thus papers and research is in greater numbers going on under different names: e.g. molecular
cybernetics -> molecular computational systems OR molecular systems theory OR molecular systemics OR
molecular information/informational systems

Please heed this when you engage in an extensive search for information to assure access to a broad range of papers.

Further reading

• Murthy, V.K and Krishnamurthy, E.V., (2009)," Multiset of Agents in a Network for Simulation of Complex
Systems", in "Recent advances in Nonlinear Dynamics and synchronization, (NDS-1) -Theory and applications,
Springer Verlag, New York,2009. Eds. K.Kyamakya et al.

• Rocha, Luis M. (1999). " Systems Modeling: Using Metaphors From Nature in Simulation and Scientific Models
(http://informatics.indiana.edu/rocha/complex/csm.htmlComplex)". BITS: Computer and Communications
News. Computing, Information, and Communications Division. Los Alamos National Laboratory. November
1999

• Ignazio Licata & Ammar Sakaji (eds) (2008). Physics of Emergence and Organization (http://www.
worldscibooks.com/physics/6692.html) , ISBN 13 978-981-277-994-6, World Scientific and Imperial College
Press.

• James S. Kim, Hyper Emotional Society (http://knol.google.eom/k/james-s-kim/21c-hyper-emotional-society/
2ycwib2vxc76q/57), Version 9. Knol. 2009 Nov 25.

External links

Articles/General Information

• Complex systems (http://www.scholarpedia.org/article/Complex_Systems) in scholarpedia.

• (European) Complex Systems Society (http://cssociety.org)

• (Australian) Complex systems research network, (http://www.complexsystems.net.au/)

• Complex Systems Modeling (http://informatics.indiana.edu/rocha/complex/csm.html) based on Luis M.
Rocha, 1999.

• CRM Complex systems research group (http://www.crm.cat/HarmonicAnalysis/defaultHarmonicAnalysis.
htm)

• Center for Complex Systems Research, Univ. of Illinois (http://www.ccsr.uiuc.edu/)

Complex Systems Biology

261

Complex Systems Biology

Complex systems biology (CSB) is a branch or subfield of mathematical and theoretical biology concerned with
complexity of both structure and function in biological organisms, as well as the emergence and evolution of
organisms and species, with emphasis being placed on the complex interactions of, and within, bionetworks , and
on the fundamental relations and relational patterns that are essential to life CSB is thus a field of

theoretical sciences aimed at discovering and modeling the relational patterns essential to life that has only a partial
overlap with complex systems theory , and also with the systems approach to biology called systems biology; this
is because the latter is restricted primarily to simplified models of biological organization and organisms, as well as

ro]

to only a general consideration of philosophical or semantic questions related to complexity in biology . Moreover,
a wide range of abstract theoretical complex systems are studied as a field of applied mathematics, with or without
relevance to biology, chemistry or physics.

External
Environment

Changing

External

Environment

.a*

— ». Complex Adaptive Behavior

-Q "an

l'£ (

^ r9e %

Changing

External

Environment

v* 1

Simple Self -Organized
Local Relationships

Changing

External

Environment

Network Representation of a Complex Adaptive System

Topics in complex systems biology

The following is only a partial list of topics covered in complex systems biology:

• Evolution theories and population genetics

• Population genetics models

• Molecular evolution theories

• Quantum biocomputation

[9]

[10] [11] [12] [13]

Quantum genetics
Relational biology

[14]

Self-reproduction (also called self-replication in a more general context)
Computational gene models

• DNA topology

• DNA sequencing theory
Evolutionary developmental biology
Autopoiesis

Protein folding

**-

TW'

"SSr

% X

J" jSp

It-- ll

^^%

.«•■— «^

Animated Molecular

Model of a DNA double

helix

Complex Systems Biology

262

Telomerase conformations and functions

Epigenetics

Interactomics

Cell signaling

Signal transduction networks

Complex neural nets

Genetic networks

Morphogenesis

Digital morphogenesis

Complex adaptive systems

Topological models of morphogenesis

Population dynamics of fisheries

Epidemiology

Telomerase

Telomerase structure and function

Related journals

Acta Biotheoretica

Bioinformatics

Biological Theory
[20]

[17]

[19]

BioSystems

Bulletin of Mathematical Biology

1221

Ecological Modelling

Journal of Mathematical Biology

Journal of Theoretical Biology

1251
Mathematical Biosciences

[26]

[21]

[23]

[21]

Medical Hypotheses

1271
Theoretical and Applied Genetics

12X1

Theoretical Biology and Medical Modelling
Theoretical Population Biology

[30]

[29]

Theory in Biosciences (formerly: Biologisches Zentralblatt)

CBS societies and institutes

• Society for Mathematical Biology

• ESMTB: European Society for Mathematical and Theoretical Biology

• Division of Mathematical Biology at NIMR ^ '

[31]

The Israeli Society for Theoretical and Mathematical Biology
Societe Francophone de Biologie Theorique

[33]

cytoplasm

- ■l l'.U.IWi

transcription nucleus

A Complex Signal Transduction
Pathway

International Society for Biosemiotic Studies

[35]

Complex Systems Biology

263

Biographies

Charles Darwin
D'Arcy Thompson
William Ross Ashby
Ludwig von Bertalanffy
Ronald Brown
Joseph Fourier
Brian Goodwin
George Karreman
Charles S. Peskin
Nicolas Rashevsky
Robert Rosen
Anatol Rapoport
Rosalind Franklin
Francis Crick
Rene Thom
Vito Volterra
Norbert Wiener

[36]

References cited

• J.D. Murray, Mathematical Biology. Springer- Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An
Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications,
2003 ISBN 0-387-95228-4.

• Thompson, DArcy W., 1992. On Growth and Form. Dover reprint of 1942, 2nd ed. (1st ed., 1917). ISBN
0-486-67135-6

• Nicolas Rashevsky. (1938)., Mathematical Biophysics. Chicago: University of Chicago Press.

• Robert Rosen. 1970. Dynamical system theory in biology. New York, Wiley-Interscience. ISBN 0-471-73550-7

• Israel, G, 2005, "Book on mathematical biology" in Grattan-Guinness, I., ed., Landmark Writings in Western
Mathematics. Elsevier: 936-44.

• Israel G (1988). "On the contribution of Volterra and Lotka to the development of modern biomathematics".
History and Philosophy of the Life Sciences 10 (1): 37-49. PMID 3045853.

• Scudo FM (March 1971). "Vito Volterra and theoretical ecology". Theoretical Population Biology 2 (1): 1—23.
doi: 10. 1016/0040-5809(71)90002-5. PMID 4950157.

• S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering.
Perseus, 2001, ISBN 0-7382-0453-6

• N.G van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN
0-444-89349-0

• I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine., Monograph, Ch.l 1 in M.
Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol. 7: 1513-1577 (1987),Pergamon Press:New York,
(updated by Hsiao Chen Lin in 2004 ISBN 0-08-036377-6

• P.G Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4

• L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6

• G Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2

• E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7

• Rosen, Robert. 1991, Life Itself: A Comprehensive Inquiry into the Nature, Origin, and Fabrication of Life,
Columbia University Press, published posthumously:

Complex Systems Biology 264

• Rosen, Robert. 2000, Essays on Life Itself, Columbia University Press.

• Rosen, Robert. 2003, "Anticipatory Systems; Philosophical, Mathematical, and Methodolical Foundations" ,
Rosen Enterprises pubis.

• S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8

• L. A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN
0-521-27477-X.

• D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3

• L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8.

• Graeme Donald Snooks, "A general theory of complex living systems: Exploring the demand side of dynamics",
Complexity, vol. 13, no. 6, July/August 2008.

• Tsekeris, Charalambos. 'Advances in Understanding Human Complex Systems", Australian Journal of Basic and
Applied Sciences, vol. 3, no. 4, October/December 2009.

• Bonner, J. T. 1988. The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University
Press.

Further reading

• A general list of Theoretical biology/Mathematical biology references, including an updated list of actively
contributing authors

rooi

• A list of references for applications of category theory in relational biology

• An updated list of publications of theoretical biologist Robert Rosen

• Theory of Biological Anthropology (Documents No. 9 and 10 in English)

• Drawing the Line Between Theoretical and Basic Biology (a forum article by Isidro T. Savillo)

[421

• Semantic Systems Biology

• Synthesis and Analysis of a Biological System , by Hiroyuki Kurata, 1999.

Notes

[I] Sprites, P; Glymour, C; Schemes, R (2000). Causation, Prediction, and Search: Adaptive Computation and Machine Learning (2nd ed.). MIT
Press.

[2] Graeme Donald Snooks, "A general theory of complex living systems: Exploring the demand side of dynamics", Complexity, vol. 13, no. 6,

July/August 2008.
[3] Bonner, J. T. 1988. The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University Press.
[4] Rosen, R.: 1958a, "A Relational Theory of Biological Systems". Bulletin of Mathematical Biophysics 20: 245-260
[5] Baianu, I. C: 2006, "Robert Rosen's Work and Complex Systems Biology", Axiomathes 16(l-2):25-34
[6] Rosen, R.: 1958b, "The Representation of Biological Systems from the Standpoint of the Theory of Categories.", Bulletin of Mathematical

Biophysics 20: 317-341.
[7] http://www.springerlink.com/index/VlRT05876H74V607.pdfI. C. Baianu, R. Brown and J. F. Glazebrook. 2007. Categorical Ontology

of Complex Spacetime Structures: The Emergence of Life and Human Consciousness., Axiomathes, 17:223—352. doi:

10.1007/sl0516-007-9011-2 .
[8] http://www.semantic-systems-biology.org/Semantic Systems Biology Portal
[9] Rosen, R. 1960. (1960). "A quantum-theoretic approach to genetic problems". Bulletin of Mathematical Biophysics 22: 227—255.

doi:10.1007/BF02478347.
[10] Rosen, R.: 1958a, "A Relational Theory of Biological Systems". Bulletin of Mathematical Biophysics 20: 245-260.

[II] Baianu, I. C: 2006, (2006). "Robert Rosen's Work and Complex Systems Biology". Axiomathes 16 (1-2): 25-34.
doi:10.1007/sl0516-005-4204-z.

[12] Rosen, R.: 1958b, (1958). "The Representation of Biological Systems from the Standpoint of the Theory of Categories". Bulletin of

Mathematical Biophysics 20: 317-341. doi:10.1007/BF02477890.
[13] http ://planetmath. org/?op=getobj &from=obj ects&id= 1 092 1
[14] "PlanetMath" (http://planetmath.org/?method=12h&from=objects&name=NaturalTransformationsOfOrganismicStructures&op=getobj).

PlanetMath. . Retrieved 2010-03-17.
[15] Faith, JJ et al. (2007). "Large-Scale Mapping and Validation of Escherichia coli Transcriptional Regulation from a Compendium of

Expression Profiles". PLoS Biology 5 (1): 54-66. doi: 10.1371/journal.pbio.0050008. PMC 1764438. PMID 17214507.

Complex Systems Biology

265

[16] Hayete, B; Gardner, TS; Collins, JJ (2007). "Size matters: network inference tackles the genome scale". Molecular Systems Biology 3: 77.
doi:10.1038/msb4100118. PMC 1828748. PMID 17299414.

[17
[18
[19

[20
[21
[22
[23
[24
[25
[26
[27
[28
[29
[30
[31
[32
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[36
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http://www.springerlink. com/link. asp?id=102835

http://bioinformatics.oupjournals.org/

http://www.mitpressjournals.org/loi/biot/

http://www.elsevier.com/locate/biosystems

http://www.springerlink.com/content/119979/

http://www.elsevier.com/locate/issn/03043800

http://www.springerlink.com/content/100436/

http://www.elsevier.com/locate/issn/0022-5193

http://www.elsevier.com/locate/mbs

http://www.harcourt-international.com/journals/mehy/

http://www.springerlink.com/content/100386/

http://www.tbiomed.com/

http ://www. elsevier. com/locate/issn/00405 809

http://www.elsevier.com/wps/product/cws_home/701802

http ://www. esmtb. org/

http://mathbio.nimr.mrc.ac.uk/wiki/Division_of_Mathematical_Biology_at_NIMR

http://bioinformatics.weizmann.ac.il/istmb/

http://www. necker. fr/sfbt/

http://www.biosemiotics.org/

http://planetphysics.org/encyclopedia/NicolasRashevsky.html

http://www.kli.ac.at/theorylab/index.html

http://planetmath.org/?method=12h&from=objects&id=10746&op=getobj

http://www.people.vcu.edu/~mikuleck/rosen.htm

http://homepage.uibk.ac.at/~c720126/humanethologie/ws/medicus/blockl/inhalt.html

http://www.scientistsolutions.com/t5844-Drawing+the+line+between+Theoretical+and+Basic+Biology.html

http://www.semantic-systems-biology.org

http://www.genome.ad.jp/manuscripts/GIW99/Poster/GIW99P66.pdf

External links

• Center for Complex Systems and Brain Sciences at Florida Atlantic (http://www.ccs.fau.edu/)

• Santa Fe Institute

• Bulletin of Mathematical Biology (http://www.springerlink.eom/content/l 19979/)

• European Society for Mathematical and Theoretical Biology (http://www.esmtb.org/)

• Journal of Mathematical Biology (http://www.springerlink.com/content/100436/)

• Biomathematics Research Centre at University of Canterbury (http://www.math.canterbury.ac.nz/bio/)

• Centre for Mathematical Biology at Oxford University (http://www.maths.ox.ac.uk/cmb/)

• Mathematical Biology at the National Institute for Medical Research (http://mathbio.nimr.mrc.ac.uk/)

• Institute for Medical BioMathematics (http://www.imbm.org/)

• Mathematical Biology Systems of Differential Equations (http://eqworld.ipmnet.ru/en/solutions/syspde/
spde-toc2.pdf) from EqWorld: The World of Mathematical Equations

• Systems Biology Workbench - a set of tools for modelling biochemical networks (http://sbw.kgi.edu)

• The Collection of Biostatistics Research Archive (http://www.biostatsresearch.com/repository/)

• Statistical Applications in Genetics and Molecular Biology (http://www.bepress.com/sagmb/)

• The International Journal of Biostatistics (http://www.bepress.com/ijb/)

• Theoretical Modeling of Cellular Physiology at Ecole Normale Superieure, Paris (http://www.biologie.ens.fr/
besmebs/)

• Theoretical and mathematical biology website (http://www.kli.ac.at/theorylab/index.html)

• Complexity Discussion Group (http://www.complex.vcu.edu/)

• UCLA Biocybernetics Laboratory (http://biocyb.cs.ucla.edu/research.html)

• TUCS Computational Biomodelling Laboratory (http://www.tucs.fi/research/labs/combio.php)

• Nagoya University Division of Biomodeling (http://www.agr.nagoya-u.ac.jp/english/e3senko-l.html)

Complex Systems Biology 266

Technische Universiteit Biomodeling and Informatics (http://www.bmi2.bmt.tue.nl/Biomedinf/)

New England Complex Systems Institute

Northwestern Institute on Complex Systems (NICO) (http://www.northwestern.edu/nico/)

Complexity Digest (http://comdig.unam.mx/)

Centra de Ciencias de la Complejidad (http://c3.fisica.unam.mx/), UNAM

Complexity Complex at the University of Warwick (http://go.warwick.ac.uk/complexity/)

Southampton Institute for Complex Systems Simulation (http://www.icss.soton.ac.uk)

Center for the Study of Complex Systems at the University of Michigan (http://www.cscs.umich.edu/)

ARC Centre for Complex Systems, Australia

(European) Complex Systems Society (http://cssociety.org)

(Australian) Complex systems research network, (http://www.complexsystems.net.au/)

Complex Systems Modeling (http://informatics.indiana.edu/rocha/complex/csm.html) based on Luis M.

Rocha, 1999.

• CRM Complex systems research group (http://www.crm.cat/HarmonicAnalysis/defaultHarmonicAnalysis.
htm)

Mathematical, Relational and Theoretical Biology

Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in
biology, medicine and biotechnology. The field may be referred to as mathematical biology or biomathematics

to stress the mathematical side, or as theoretical biology to stress the biological side. It includes at least four
major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB),
bioinformatics and computational biomodeling/biocomputing. Mathematical biology aims at the mathematical

representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques
and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. For
example, in cell biology, protein interactions are often represented as "cartoon" models, which, although easy to
visualize, do not accurately describe the systems studied. In order to do this, precise mathematical models are
required. By describing the systems in a quantitative manner, their behavior can be better simulated, and hence
properties can be predicted that might not be evident to the experimenter.

Such mathematicial areas as calculus, probability theory, statistics, linear algebra, abstract algebra, graph theory,
combinatorics, algebraic geometry, topology, dynamical systems, differential equations and coding theory are now
being applied in biology.

Importance

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the
field. Some reasons for this include:

• the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand
without the use of analytical tools,

• recent development of mathematical tools such as chaos theory to help understand complex, nonlinear
mechanisms in biology,

• an increase in computing power which enables calculations and simulations to be performed that were not
previously possible, and

• an increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other
complications involved in human and animal research.

Mathematical, Relational and Theoretical Biology 267

Areas of research

Several areas of specialized research in mathematical and theoretical biology as well as external links

to related projects in various universities are concisely presented in the following subsections, including also a large
number of appropriate validating references from a list of several thousands of published authors contributing to this
field. Many of the included examples are characterised by highly complex, nonlinear, and supercomplex
mechanisms, as it is being increasingly recognised that the result of such interactions may only be understood
through a combination of mathematical, logical, physical/chemical, molecular and computational models. Due to the
wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between
mathematicians, biomathematicians, theoretical biologists, physicists, biophysicists, biochemists, bioengineers,
engineers, biologists, physiologists, research physicians, biomedical researchers, oncologists, molecular biologists,
geneticists, embryologists, zoologists, chemists, etc.

Computer models and automata theory

A monograph on this topic summarizes an extensive amount of published research in this area up to 1987,

including subsections in the following areas: computer modeling in biology and medicine, arterial system models,

ri2i
neuron models, biochemical and oscillation networks, quantum automata , quantum computers in molecular

biology and genetics, cancer modelling, neural nets, genetic networks, abstract relational biology,

metabolic-replication systems, category theory applications in biology and medicine, automata theory, cellular

automata, tessallation models and complete self-reproduction , chaotic systems in organisms, relational

biology and organismic theories. This published report also includes 390 references to peer-reviewed articles

by a large number of authors.

Modeling cell and molecular biology

T91
This area has received a boost due to the growing importance of molecular biology.

T221
Mechanics of biological tissues

Theoretical enzymology and enzyme kinetics

Cancer modelling and simulation

T251

Modelling the movement of interacting cell populations
Mathematical modelling of scar tissue formation

T271

Mathematical modelling of intracellular dynamics

T2R1

Mathematical modelling of the cell cycle
Modelling physiological systems

[291

• Modelling of arterial disease

• Multi-scale modelling of the heart

Molecular set theory

Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical

nil
biology and especially in Mathematical Medicine. Molecular set theory (MST) is a mathematical formulation of

the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical

transformations represented by set-theoretical mappings between molecular sets. In a more general sense, MST is the

theory of molecular categories defined as categories of molecular sets and their chemical transformations represented

as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of

clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to

nil T321
Physiology, Clinical Biochemistry and Medicine.

Mathematical, Relational and Theoretical Biology 268

Population dynamics

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back
to the 19th century. The Lotka— Volterra predator-prey equations are a famous example. In the past 30 years,
population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith.
Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population
dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the
study of infectious disease affecting populations. Various models of the spread of infections have been proposed and
analyzed, and provide important results that may be applied to health policy decisions.

Mathematical methods

A model of a biological system is converted into a system of equations, although the word 'model' is often used
synonymously with the system of corresponding equations. The solution of the equations, by either analytical or
numerical means, describes how the biological system behaves either over time or at equilibrium. There are many
different types of equations and the type of behavior that can occur is dependent on both the model and the equations
used. The model often makes assumptions about the system. The equations may also make assumptions about the
nature of what may occur.

Mathematical biophysics

The earlier stages of mathematical biology were dominated by mathematical biophysics, described as the application
of mathematics in biophysics, often involving specific physical/mathematical models of biosystems and their
components or compartments.

The following is a list of mathematical descriptions and their assumptions.

Deterministic processes (dynamical systems)

A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in
time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.

• Difference equations/Maps — discrete time, continuous state space.

• Ordinary differential equations — continuous time, continuous state space, no spatial derivatives. See also:
Numerical ordinary differential equations.

• Partial differential equations — continuous time, continuous state space, spatial derivatives. See also: Numerical
partial differential equations.

Stochastic processes (random dynamical systems)

A random mapping between an initial state and a final state, making the state of the system a random variable with a
corresponding probability distribution.

• Non-Markovian processes — generalized master equation — continuous time with memory of past events, discrete
state space, waiting times of events (or transitions between states) discretely occur and have a generalized
probability distribution.

• Jump Markov process — master equation — continuous time with no memory of past events, discrete state space,
waiting times between events discretely occur and are exponentially distributed. See also: Monte Carlo method
for numerical simulation methods, specifically dynamic Monte Carlo method and Gillespie algorithm.

• Continuous Markov process — stochastic differential equations or a Fokker-Planck equation — continuous time,
continuous state space, events occur continuously according to a random Wiener process.

Spatial modelling

One classic work in this area is Alan Turing's paper on morphogenesis entitled The Chemical Basis of
Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society.

Mathematical, Relational and Theoretical Biology 269

T331

• Travelling waves in a wound-healing assay

• Swarming behaviour

• A mechanochemical theory of morphogenesis

T371
Spatial distribution modeling using plot samples

• Biological pattern formation

• Spatial distribut

Relational biology

HOI

Abstract Relational Biology (ARB) is concerned with the study of general, relational models of complex
biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest
models in ARB are the Metabolic-Replication, or (M,R)~ systems introduced by Robert Rosen in 1957-1958 as
abstract, relational models of cellular and organismal organization.

Phylogenetics

Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and

[391
networks based on inherited characteristics

Model example: the cell cycle

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to
cancers. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid
results. Two research groups have produced several models of the cell cycle simulating several organisms.

They have recently produced a generic eukaryotic cell cycle model which can represent a particular eukaryote
depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due
to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et
al., 2006).

By means of a system of ordinary differential equations these models show the change in time (dynamical system) of
the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model
describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process).
To obtain these equations an iterative series of steps must be done: first the several models and observations are
combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential
equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme substrate
reactions and Goldbeter— Koshland kinetics for ultrasensitive transcription factors, afterwards the parameters of the
equations (rate constants, enzyme efficiency coefficients and Michealis constants) must be fitted to match
observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring
diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such
as protein half-life and cell size.

In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or
by analysis.

In a simulation, given a starting vector (list of the values of the variables), the progression of the system is calculated
by solving the equations at each time-frame in small increments.

Mathematical, Relational and Theoretical Biology

270

BIFURCATION DIAGRAM

Fixed Points

Stable steady-state:

Saddle steady-state:

System is in an exitatory phase indipendent of mass because

ustable steady- stales repeil (one or more positive eigenvalues)

• o Stable/Unstable limit cycle max/min:

The system is in a loop, so at that mass the [MPF] will oscillate

SN1

SN2

Saddle Node:

.instable steady-states annihilate, beyond
10 equilibrium points: these bifurcation
rthe exit from G1 and G2 respectively

SN1 1 2

cell mass (an.)

dMass/dt = kg row tfyMass (exponential growth]

d[Cln2]/dt = (k si + k S 2'[SBF])-mass - k„- [Cln2]

The parameter mass directly controls cyclin levels, expressing

implicitly its yet unknown mass dependant control mechanism

MB Hopf Bifurcation

A stable and an unstahle steady-states annihilate resulting in
an unstable limit cycle (eigenvalues have no Real part)

SNIPER SNIPER Bifurcation

merges from a stable and

In analysis, the proprieties of the

equations are used to investigate the

behavior of the system depending of

the values of the parameters and

variables. A system of differential

equations can be represented as a

vector field, where each vector

described the change (in concentration

of two or more protein) determining

where and how fast the trajectory

(simulation) is heading. Vector fields

can have several special points: a

stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an

unstable point, either a source or a saddle point which repels (forcing the concentrations to change away from a

certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the

concentrations oscillate).

A better representation which can handle the large number of variables and parameters is called a bifurcation

diagram (Bifurcation theory): the presence of these special steady-state points at certain values of a parameter (e.g.

mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a

bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the

cell cycle has phases (partially corresponding to Gl and G2) in which mass, via a stable point, controls cyclin levels,

and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a

bifurcation event (Cell cycle checkpoint), the system cannot go back to the previous levels since at the current mass

the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a

checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a

Hopf bifurcation and an infinite period bifurcation.

Notes

[I] Mathematical and Theoretical Biology: A European Perspective (http://sciencecareers.sciencemag.org/career_development/
previous_issues/articles/2870/mathematical_and_theoretical_biology_a_european_perspective)

[2] "There is a subtle difference between mathematical biologists and theoretical biologists. Mathematical biologists tend to be employed in

mathematical departments and to be a bit more interested in math inspired by biology than in the biological problems themselves, and vice

versa." Careers in theoretical biology (http://life.biology.mcmaster.ca/~brian/biomath/careers.theo.biol.html)
[3] Baianu, I. C. 1987, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.) Mathematical Models in Medicine,

vol. 7., Ch.ll Pergamon Press, New York, 1513-1577. http://cogprints.org/3687/
[4] http://library.bjcancer.org/ebook/109.pdf L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN

1-58488-361-8.
[5] Robeva, Raina; et al (Fall 2010). "Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics".

CBELife Sciences Education (The American Society for Cell Biology) 9 (3): 227-240. doi:10.1187/cbe.lO-03-0019. PMC 2931670.

PMID 20810955.
[6] Baianu, I. C; Brown, R.; Georgescu, G.; Glazebrook, J. F. (2006). "Complex Non-linear Biodynamics in Categories, Higher Dimensional

Algebra and Lukasiewicz— Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks". Axiomathes 16: 65.

doi:10.1007/sl0516-005-3973-8.
[7] (http://en.scientificcommons.org/1857371)
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[9] "Research in Mathematical Biology" (http://www.maths.gla.ac.uk/research/groups/biology/kal.htm). Maths.gla.ac.uk. . Retrieved

2008-09-10.
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/acube.org/volume_23/v23-lpl l-36.pdf) New Link (Aug 2010) (http://papa.indstate.edu/amcbt/volume_23/v23-lpll-36.pdf)

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Mathematical, Relational and Theoretical Biology 271

[12] http://planetphysics.org/encyclopedia/QuantumAutomaton.html

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• Mangel, M. 1990. Special Issue, Classics of Theoretical Biology (part 1). Bull. Math. Biol. 52(1/2): 1-318.

• Mangel, M. 2006. The Theoretical Biologist's Toolbox. Quantitative Methods for Ecology and Evolutionary
Biology. Cambridge University Press.

• Prusinkiewicz, P. & Lindenmeyer, A. 1990. The Algorithmic Beauty of Plants. Berlin: Springer-Verlag.

• Reinke, J. 1901. Einleitung in die theoretische Biologic Berlin: Verlag von Gebriider Paetel.

• Thompson, D.W. 1942. On Growth and Form. 2nd ed. Cambridge: Cambridge University Press: 2. vols.

• Uexkiill, J. v. 1920. Theoretische Biologic Berlin: Gebr. Paetel.

• Vogel, S. 1988. Life's Devices: The Physical World of Animals and Plants. Princeton: Princeton University Press.

• Waddington, C.H. 1968-1972. Towards a Theoretical Biology. 4 vols. Edinburg: Edinburg University Press.

Mathematical, Relational and Theoretical Biology 273

Further reading

• Hoppensteadt, F. (September 1995). "Getting Started in Mathematical Biology" (http://www.ams.org/notices/
199509/hoppensteadt.pdf). Notices of American Mathematical Society.

• Reed, M. C. (March 2004). "Why Is Mathematical Biology So Hard?" (http://www.resnet.wm.edu/~jxshix/
math490/reed.pdf). Notices of American Mathematical Society.

• May, R. M. (2004). "Uses and Abuses of Mathematics in Biology". Science 303 (5659): 790-793.
doi: 10. 1126/science. 1094442. PMID 14764866.

• Murray, J. D. (1988). "How the leopard gets its spots?" (http://www.resnet.wm.edu/~jxshix/math490/murray.
doc). Scientific American 258 (3): 80—87. doi:10.1038/scientificamerican0388-80.

• Schnell, S.; Grima, R.; Maini, P. K. (2007). "Multiscale Modeling in Biology" (http://eprints.maths.ox.ac.uk/
567/01/224.pdf). American Scientist 95: 134-142.

• Chen, Katherine C; Calzone, Laurence; Csikasz-Nagy, Attila; Cross, FR; Cross, Frederick R.; Novak, Bela;
Tyson, John J. (2004). "Integrative analysis of cell cycle control in budding yeast". Mol Biol Cell 15 (8):
3841-3862. doi:10.1091/mbc.E03-l 1-0794. PMC 491841. PMID 15169868.

• Csikasz-Nagy, Attila; Battogtokh, Dorjsuren; Chen, Katherine C; Novak, Bela; Tyson, John J. (2006). "Analysis
of a generic model of eukaryotic cell-cycle regulation". Biophys J. 90 (12): 4361—4379.

doi: 10. 1529/biophysj. 106.08 1240. PMC 1471857. PMID 16581849.

• Fuss, H; Dubitzky, Werner; Downes, C. Stephen; Kurth, Mary Jo (2005). "Mathematical models of cell cycle
regulation". Brief Bioinform. 6 (2): 163-177. doi:10.1093/bib/6.2.163. PMID 15975225.

• Lovrics, Anna; Csikasz-Nagy, Attila; Zselyl, Istvan Gy; Zador, Judit; Turanyi, Tamas; Novak, Bela (2006).
"Time scale and dimension analysis of a budding yeast cell cycle model". BMC Bioinform. 9 (7): 494.

doi: 10. 11 86/147 1-2 105-7-494.

External links

The Society for Mathematical Biology (http://www.smb.org/)

Theoretical and mathematical biology website (http://www.kli.ac.at/theorylab/index.html)

Complexity Discussion Group (http://www.complex.vcu.edu/)

UCLA Biocybernetics Laboratory (http://biocyb.cs.ucla.edu/research.html)

TUCS Computational Biomodelling Laboratory (http://www.tucs.fi/research/labs/combio.php)

Nagoya University Division of Biomodeling (http://www.agr.nagoya-u.ac.jp/english/e3senko-l.html)

Technische Universiteit Biomodeling and Informatics (http://www.bmi2.bmt.tue.nl/Biomedinf/)

BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology (http://wiki.

biological-cybernetics.de)

Bulletin of Mathematical Biology (http://www.springerlink.com/content/119979/)

European Society for Mathematical and Theoretical Biology (http://www.esmtb.org/)

Journal of Mathematical Biology (http://www.springerlink.com/content/100436/)

Biomathematics Research Centre at University of Canterbury (http://www.math.canterbury.ac.nz/bio/)

Centre for Mathematical Biology at Oxford University (http://www.maths.ox.ac.uk/cmb/)

Mathematical Biology at the National Institute for Medical Research (http://mathbio.nimr.mrc.ac.uk/)

Institute for Medical BioMathematics (http://www.imbm.org/)

Mathematical Biology Systems of Differential Equations (http://eqworld.ipmnet.ru/en/solutions/syspde/

spde-toc2.pdf) from EqWorld: The World of Mathematical Equations

Systems Biology Workbench - a set of tools for modelling biochemical networks (http://sbw.kgi.edu)

The Collection of Biostatistics Research Archive (http://www.biostatsresearch.com/repository/)

Statistical Applications in Genetics and Molecular Biology (http://www.bepress.com/sagmb/)

The International Journal of Biostatistics (http://www.bepress.com/ijb/)

Mathematical, Relational and Theoretical Biology 274

• Theoretical Modeling of Cellular Physiology at Ecole Normale Superieure, Paris (http://www.biologie.ens.fr/
bcsmcbs/)

Lists of references

• A general list of Theoretical biology/Mathematical biology references, including an updated list of actively
contributing authors (http://www.kli.ac.at/theorylab/index.html).

• A list of references for applications of category theory in relational biology (http://planetmath.org/
?method=12h&from=objects&id=10746&op=getobj).

• An updated list of publications of theoretical biologist Robert Rosen (http://www.people.vcu.edu/~mikuleck/
rosen.htm)

• Theory of Biological Anthropology (Documents No. 9 and 10 in English) (http://homepage.uibk.ac.at/
~c720126/humanethologie/ws/medicus/blockl/inhalt.html)

• Drawing the Line Between Theoretical and Basic Biology (a forum article by Isidro T. Savillo) (http://www.
scientistsolutions. com/ t5844-Drawing+the+line+between+Theoretical+and+Basic+Biology. html)

Related journals

Acta Biotheoretica (http://www . springerlink. com/link. asp?id= 1 02835)

B ioinformatics (http ://bioinformatics . oupjournals . org/)

Biological Theory (http://www.mitpressjournals.org/loi/biot/)

BioSystems (http://www.elsevier.com/locate/biosystems)

Bulletin of Mathematical Biology (http://www.springerlink.com/content/119979/)

Ecological Modelling (http://www.elsevier.com/locate/issn/03043800)

Journal of Mathematical Biology (http://www.springerlink.com/content/100436/)

Journal of Theoretical Biology (http://www.elsevier.com/locate/issn/0022-5193)

Journal of the Royal Society Interface (http://publishing.royalsociety.org/index.cfm?page=1058#)

Mathematical Biosciences (http://www.elsevier.com/locate/mbs)

Medical Hypotheses (http://www.harcourt-international.com/journals/mehy/)

Rivista di Biologia-Biology Forum (http://www.tilgher.it/biologiae.html)

Theoretical and Applied Genetics (http://www.springerlink.com/content/100386/)

Theoretical Biology and Medical Modelling (http://www.tbiomed.com/)

Theoretical Population Biology (http://www.elsevier.com/locate/issn/00405809)

Theory in Biosciences (http://www.elsevier.com/wps/product/cws_home/701802) (formerly: Biologisches

Zentralblatt)

Related societies

• ESMTB: European Society for Mathematical and Theoretical Biology (http://www.esmtb.org/)

• The Israeli Society for Theoretical and Mathematical Biology (http://bioinformatics.weizmann.ac.il/istmb/)

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• International Society for Biosemiotic Studies (http://www.biosemiotics.org/)

Lotka— Volterra Differential Equations in Population Biology 275

Lotka-Volterra Differential Equations in
Population Biology

The Lotka— Volterra equations, also known as the predator— prey equations, are a pair of first-order, non-linear,
differential equations frequently used to describe the dynamics of biological systems in which two species interact,
one a predator and one its prey. They evolve in time according to the pair of equations:

— = x{a - (3y)

M = -vh ~ 5x )

where,

• y is the number of some predator (for example, wolves);

• x is the number of its prey (for example, rabbits);

di/ dx

• _and — represent the growth of the two populations against time;

dt dt

• t represents the time; and

• a,j3,y and 6 are parameters representing the interaction of the two species.

The Lotka— Volterra system of equations is an example of a Kolmogorov model, which is a more general

framework that can model the dynamics of ecological systems with predator-prey interactions, competition, disease,
and mutualism.

History

The Lotka— Volterra predator— prey model was initially proposed by Alfred J. Lotka "in the theory of autocatalytic
chemical reactions" in 1910. This was effectively the logistic equation, which was originally derived by

Pierre Francois Verhulst. In 1920 Lotka extended, via Kolmogorov (see above), the model to "organic systems"

roi

using a plant species and a herbivorous animal species as an example and in 1925 he utilised the equations to
analyse predator-prey interactions in his book on biomathematics arriving at the equations that we know today.
Vito Volterra, who made a statistical analysis of fish catches in the Adriatic independently investigated the
equations in 1926.

C.S. Holling extended this model yet again, in two 1959 papers, in which he proposed the idea of functional

ri2i ri3i
response. Both the Lotka-Volterra model and Holling's extensions have been used to model the moose and

wolf populations in Isle Royale National Park, which with over 50 published papers is one of the best studied

predator-prey relationships.

Lotka— Volterra Differential Equations in Population Biology 276

In economics

The Lotka— Volterra equations have a long history of use in economic theory; their initial application is commonly
credited to Richard Goodwin in 1965 or 1967. In economics, links are between many if not all industries; a

proposed way to model the dynamics of various industries has been by introducing trophic functions between
various sectors, and ignoring smaller sectors by considering the interactions of only two industrial sectors.

Physical meanings of the equations

The Lotka-Volterra model makes a number of assumptions about the environment and evolution of the predator and
prey populations:

1. The prey population finds ample food at all times.

2. The food supply of the predator population depends entirely on the prey populations.

3. The rate of change of population is proportional to its size.

4. During the process, the environment does not change in favour of one species and the genetic adaptation is
sufficiently slow.

As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the
generations of both the predator and prey are continually overlapping.

Prey

When multiplied out, the prey equation becomes:

— = ax — pxy.
dt H y

The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation;
this exponential growth is represented in the equation above by the term ax. The rate of predation upon the prey is
assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by j3xy. If
either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own
growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes:

dy .

— =dxy- 7y.

In this equation, Sxy represents the growth of the predator population. (Note the similarity to the predation rate;
however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the
rate at which it consumes the prey), yy represents the loss rate of the predators due to either natural death or
emigration; it leads to an exponential decay in the absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus
natural death.

Lotka— Volterra Differential Equations in Population Biology

277

Solutions to the equations

The equations have periodic solutions which do not have a simple expression in terms of the usual trigonometric
functions. However, a linearization of the equations yields a solution similar to simple harmonic motion with the
population of predators following that of prey by 90°.

c
o

a
o

■prey
■predators

time

An example problem

Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80
baboons and 40 cheetahs, one can plot the progression of the two species over time. The choice of time interval is
arbitrary.

160

140

0_
"O

120

100

° 60

/ 1

r-^rey (uauoons)

PrbLklurafufibuldfi

LZ 1

■

.LjSs^

/\^H

^ V_

5 10

Time (dimension less)

One can also plot a solution which corresponds to the oscillatory nature of the population of the two species. This
solution is in a state of dynamic equilibrium. At any given time in this phase plane, the system is in a limit cycle and

Lotka— Volterra Differential Equations in Population Biology

278

lies somewhere on the inside of these elliptical solutions. There is no particular requirement on the system to begin
within a limit cycle and thus in a stable solution, however, it will always reach one eventually.

60 80 100 120 140

Number of Prey (baboons)

200

These graphs clearly illustrate a serious problem with this as a biological model: in each cycle, the baboon
population is reduced to extremely low numbers yet recovers (while the cheetah population remains sizeable at the
lowest baboon density). Given chance fluctuations, discrete numbers of individuals, and the family structure and
lifecycle of baboons, the baboons actually go extinct and by consequence the cheetahs as well. This modelling

T221 —1 8

problem has been called the "atto-fox problem", an atto-fox being an imaginary 10 of a fox, in relation to
rabies modelling in the UK.

Dynamics of the system

In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and
decline. As the predator population is low the prey population will increase again. These dynamics continue in a
cycle of growth and decline.

Population equilibrium

Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the
derivatives are equal to 0.

x(pt — (3y) —
-y{i -6x) =

When solved for x and y the above system of equations yields
and

Lotka— Volterra Differential Equations in Population Biology

279

y =

7
> X= 6

Hence, there are two equilibria.

The first solution effectively represents the extinction of both species. If both populations are at 0, then they will
continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their
current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this
equilibrium is achieved depend on the chosen values of the parameters, a, |3, y, and 5.

Stability of the fixed points

The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives,
while the other fixed point requires a slightly more sophisticated method.

The Jacobian matrix of the predator-prey model is

J(x, y)

a — /3y —f3x
5y 8x — 7

First fixed point

When evaluated at the steady state of (0, 0) the Jacobian matrix / becomes

J(0, 0)

-7

The eigenvalues of this matrix are

Ai = a, A 2 = -7.
In the model a and y are always greater than zero, and as such the sign of the eigenvalues above will always differ.
Hence the fixed point at the origin is a saddle point.

The stability of this fixed point is of importance. If it were stable, non-zero populations might be attracted towards it,
and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial
population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, we find that the
extinction of both species is difficult in the model. (In fact, this can only occur if the prey are artificially completely
eradicated, causing the predators to die of starvation. If the predators are eradicated, the prey population grows
without bound in this simple model).

Second fixed point

Evaluating / at the second fixed point we get

7 a

The eigenvalues of this matrix are

X 1 = i^/cry, A 2 = —2-^/07.
As the eigenvalues are both purely imaginary, this fixed point is not hyperbolic, so no conclusions can be drawn
from the linear analysis. However, the system admits a constant of motion

y a e- py x^e- Sx ,

and the level curves, where K = const, are closed trajectories surrounding the fixed point. Consequently, the levels of
the predator and prey populations cycle, and oscillate around this fixed point.

The largest value of the constant K can be obtained by solving the optimization problem

Lotka— Volterra Differential Equations in Population Biology 280

y <* e -Py x i e -6* = JLfL , max .

e &x+fjy x,y>0

(l a \
The maximal value of K is attained at the stationary point I — , — I and it is given by

where e is Euler's Number.

Notes

[I] Freedman, H.I., Deterministic Mathematical Models in Population Ecology, Marcel Dekker, (1980)

[2] Brauer, F. and Castillo-Chavez, C, Mathematical Models in Population Biology and Epidemiology, Springer- Verlag, (2000)

[3] Hoppensteadt, F., "Predator-prey model" (http://www.scholarpedia.org/article/Predator-prey_model), Scholarpedia, 1(10), 1563, (2006)

[4] Lotka, A.J., "Contribution to the Theory of Periodic Reaction", J. Phys. Chem., 14 (3), pp 271-274 (1910)

[5] Goel, N.S. et al., "On the Volterra and Other Non-Linear Models of Interacting Populations", Academic Press Inc., (1971)

[6] Berryman, A. A., "The Origins and Evolution of Predator-Prey Theory" (http://entomology.wsu.edu/profiles/06BerrymanWeb/

Berryman(92)Origins.pdf), Ecology, 73(5), 1530-1535, (1992)
[7] Verhulst, P.H., "Notice sur la loi que la population poursuit dans son accroissement" (http://books. google. com/books ?hl=fr&

id=8GsEAAAAYAAJ&jtp=113#v=onepage&q=&f=false). Corresp. mathe'matique et physique 10, 113-121, (1838)
[8] Lotka, A.J., "Analytical Note on Certain Rhythmic Relations in Organic Systems" (http://www.ncbi.nlm.nih.gov/pmc/articles/

PMC1084562/pdf/pnas01916-0016.pdf), Proc. Natl. Acad. Sci. U.S., 6, 410-415, (1920)
[9] Lotka, A.J., Elements of Physical Biology, Williams and Wilkins, (1925)
[10] Volterra, V., "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi", Mem. Acad. Lincei Roma, 2, 31-113, (1926)

[II] Volterra, V., Variations and fluctuations of the number of individuals in animal species living together in Animal Ecology, Chapman, R.N.
(ed), McGraw-Hill, (1931)

[12] Holling, C.S., "The components of predation as revealed by a study of small mammal predation of the European Pine Sawfly", Can. Ent, 91,

293-320, (1959a)
[13] Holling, C.S., "Some characteristics of simple types of predation and parasitism", Can. Ent, 91, 385-398, (1959b)
[14] Jost, C, Devulder, G., Vucetich, J. A., Peterson, R., and Arditi, R., "The wolves of Isle Royale display scale-invariant satiation and density

dependent predation on moose" (http://www.isleroyalewolf.org/techpubs/techpubs/ISROpubs_files/Jost et al.pdf), J. Anim. Ecol, 74(5),

809-816 (2005)
[15] Gandolfo, G., "Giuseppe Palomba and the Lotka- Volterra equations", Rendiconti Lincei, 19(4), 347-257, (2008)
[16] Goodwin, R.M. , "A Growth Cycle", Socialism, Capitalism and Economic Growth, Feinstein, C.H. (ed.), Cambridge University Press,

(1967)
[17] Desai, M. and Ormerod, P. "Richard Goodwin: A Short Appreciation" (http://www.paulormerod.com/pdf/economicjournall998.pdf),

The Economic Journal, 108(450), 1431-1435 (1998)
[18] Nasritdinov, G. and Dalimov, R.T., "Limit cycle, trophic function and the dynamics of intersectoral interaction" (http://www.maxwellsci.

com/print/crjet/v2-32-40.pdf), Current Research J. of Economic Theory, 2(2), 32-40, (2010)
[19] Haken, H., Synergetics: introduction and advanced topics (http://books. google. co.uk/books?hl=en&lr=&id=0bc6cLK0w7YC&

oi=fnd&pg=PAl&dq=Haken,+Synergetics,+Springer- Verlag, +2004&ots=dw6IClrdLp&

sig=_IIHtwt-Lxsi4694w2jFUhbGWoY#v=onepage&q=Haken, Synergetics, Springer- Verlag, 2004&f=false), Springer- Verlag, (2004)
[20] Cooke, D. and Hiorns, R.W. et al., The Mathematical Theory of the Dynamics of Biological Populations II, Academic Press Inc., (1981)
[21] Tong, H., Threshold Models in Non-linear Time Series Analysis, Springer— Verlag, (1983)
[22] Mollison, D., "Dependence of epidemic and population velocities on basic parameters" (ftp://ftp.ma.hw.ac.uk/pub/denis/velocities.

pdf), Math. Biosci, 107, 255-287, (1991)

Lotka— Volterra Differential Equations in Population Biology 281

References

• E. R. Leigh (1968) The ecological role of Volterra's equations, in Some Mathematical Problems in Biology — a
modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903.

• Understanding Nonlinear Dynamics. Daniel Kaplan and Leon Glass.

• J.D. Murray. Mathematical Biology I: An Introduction. Springer- Verlag, 2003

External links

• Lotka— Volterra Predator-Prey Model (http://www.egwald.ca/nonlineardynamics/twodimensionaldynamics.
php#predatorpreymodel) by Elmer G. Wiens

• Lotka- Volterra Model (http://math.fullerton.edu/mathews/n2003/Lotka-VolterraMod.html)

• NANIA Lotka-Volterra applet (http://www.ph.ed.ac.uk/nania/lv/lv.html)

• Lotka Algorithmic Simulation (http://jseed.sourceforge.net/lotka/index.html) Similar program, in Javascript
(requires an HTML5 browser).

282

Chaotic Dynamics

Chaos theory

Chaos theory is a field of study in applied
mathematics, with applications in several
disciplines including physics, economics,
biology, and philosophy. Chaos theory
studies the behavior of dynamical systems
that are highly sensitive to initial conditions;
an effect which is popularly referred to as
the butterfly effect. Small differences in
initial conditions (such as those due to
rounding errors in numerical computation)
yield widely diverging outcomes for chaotic
systems, rendering long-term prediction
impossible in general. This happens even
though these systems are deterministic,
meaning that their future behavior is fully

determined by their initial conditions, with

121
no random elements involved. In other

the deterministic nature of these

[3]

words

systems does not make them predictable

[4]

This behavior is known as deterministic

A plot of the Lorenz attractor for values r = 28, o = 10, b = 8/3.

chaos, or simply chaos.

Chaotic behavior can be observed in many natural systems, such as the weather. Explanation of such behavior may
be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence
plots and Poincare maps.

Applications

A conus textile shell, similar in appearance to

Rule 30, a cellular automaton with chaotic

behaviour.

Chaos theory is applied in many scientific disciplines: mathematics,
programming, microbiology, biology, computer science, economics,

engineering, finance, philosophy, physics, politics,

population dynamics, psychology, robotics, and meteorology.

Chaotic behavior has been observed in the laboratory in a variety of
systems including electrical circuits, lasers, oscillating chemical
reactions, fluid dynamics, and mechanical and magneto-mechanical
devices, as well as computer models of chaotic processes. Observations
of chaotic behavior in nature include changes in weather, the
dynamics of satellites in the solar system, the time evolution of the

Chaos theory

283

magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons,
and molecular vibrations. There is some controversy over the existence of chaotic dynamics in plate tectonics and in

• [13] [14] [15]

economics.

A successful application of chaos theory is in ecology where dynamical systems such as the Ricker model have been
used to show how population growth under density dependence can lead to chaotic dynamics.

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of
seemingly random seizures by observing initial conditions

[16]

Quantum chaos theory studies how the correspondence between quantum mechanics and classical mechanics works

ri7i nsi

in the context of chaotic systems. Recently, another field, called relativistic chaos, has emerged to describe

systems that follow the laws of general relativity.

[191
The motion of N stars in response to their self-gravity (the gravitational iV-body problem) is generically chaotic.

In electrical engineering, chaotic systems are used in communications, random number generators, and encryption
systems.

In numerical analysis, the Newton-Raphson method of approximating the roots of a function can lead to chaotic
iterations if the function has no real roots

[20]

Chaotic dynamics

[21]

In common usage, "chaos" means "a state of disorder".
However, in chaos theory, the term is defined more precisely.
Although there is no universally accepted mathematical definition
of chaos, a commonly used definition says that, for a dynamical
system to be classified as chaotic, it must have the following

^- P2]

properties:

1. it must be sensitive to initial conditions;

2. it must be topologically mixing; and

3. its periodic orbits must be dense.

The requirement for sensitive dependence on initial conditions
implies that there is a set of initial conditions of positive measure
which do not converge to a cycle of any length.

Sensitivity to initial conditions

The map defined by x — > 4 x (1 — x) and y — > x + y if x

+ )' < 1 (x + y — 1 otherwise) displays sensitivity to

initial conditions. Here two series of x and v values

diverge markedly over time from a tiny initial

difference.

Sensitivity to initial conditions means that each point in such a
system is arbitrarily closely approximated by other points with
significantly different future trajectories. Thus, an arbitrarily small
perturbation of the current trajectory may lead to significantly
different future behaviour. However, it has been shown that the last two properties in the list above actually imply

[23] [24]

sensitivity to initial conditions and if attention is restricted to intervals, the second property implies the other

[25]

two (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above
list ). It is interesting that the most practically significant condition, that of sensitivity to initial conditions, is
actually redundant in the definition, being implied by two (or for intervals, one) purely topological conditions, which
are therefore of greater interest to mathematicians.

Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper
given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C.
entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas? The flapping wing

Chaos theory

284

represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale
phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the

system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable. This

1271
is most familiar in the case of weather, which is generally predictable only about a week ahead.

The Lyapunov exponent characterises the extent of the sensitivity to initial conditions. Quantitatively, two
trajectories in phase space with initial separation <5Zo diverge

\SZ(t)\ ^e M \8Z \
where X is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial
separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the
number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov
exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as
an indication that the system is chaotic.

There are also measure-theoretic mathematical conditions (discussed in ergodic theory) such as mixing or being a
K-system which relate to sensitivity of initial conditions and chaos

Topological mixing

Topological mixing (or topological transitivity) means that the
system will evolve over time so that any given region or open set
of its phase space will eventually overlap with any other given
region. This mathematical concept of "mixing" corresponds to the
standard intuition, and the mixing of colored dyes or fluids is an
example of a chaotic system.

Topological mixing is often omitted from popular accounts of
chaos, which equate chaos with sensitivity to initial conditions.
However, sensitive dependence on initial conditions alone does
not give chaos. For example, consider the simple dynamical
system produced by repeatedly doubling an initial value. This
system has sensitive dependence on initial conditions everywhere,
since any pair of nearby points will eventually become widely
separated. However, this example has no topological mixing, and
therefore has no chaos. Indeed, it has extremely simple behaviour:
all points except tend to infinity.

The map defined by x — > 4 je (1 — x) and y — > x + v if x
+ y < 1 (x + y — 1 otherwise) also displays topological

mixing. Here the blue region is transformed by the

dynamics first to the purple region, then to the pink and

red regions, and eventually to a cloud of points

scattered across the space.

Density of periodic orbits

Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.
Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may
not be chaotic. For example, an irrational rotation of the circle is topologically transitive, but does not have dense

no]

periodic orbits, and hence does not have sensitive dependence on initial conditions. The one-dimensional logistic

map defined by x — > 4 x (1 — x) is one of the simplest systems with density of periodic orbits. For example, 5 ~ v5 _>

5+v| _» 5rr^l(or approximately 0.3454915

8 8

0.9045085 -» 0.3454915) is an (unstable) orbit of period 2, and

similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem)

[29]

Chaos theory

285

Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which
exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely
chaotic orbits.

Strange attractors

Some dynamical systems, like the
one-dimensional logistic map defined by x
— > 4 x (1 — x), are chaotic everywhere, but
in many cases chaotic behaviour is found
only in a subset of phase space. The cases of
most interest arise when the chaotic
behaviour takes place on an attractor, since
then a large set of initial conditions will lead
to orbits that converge to this chaotic region.

An easy way to visualize a chaotic attractor

is to start with a point in the basin of

attraction of the attractor, and then simply

plot its subsequent orbit. Because of the

topological transitivity condition, this is

likely to produce a picture of the entire final

attractor, and indeed both orbits shown in

the figure on the right give a picture of the

general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz

weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it

was not only one of the first, but it is also one of the most complex and as such gives rise to a very interesting pattern

which looks like the wings of a butterfly.

Unlike fixed-point attractors and limit cycles, the attractors which arise from chaotic systems, known as strange
attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as
the Lorenz system) and in some discrete systems (such as the Henon map). Other discrete dynamical systems have a
repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points — Julia
sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and
a fractal dimension can be calculated for them.

The Lorenz attractor displays chaotic behavior. These two plots demonstrate

sensitive dependence on initial conditions within the region of phase space

occupied by the attractor.

Chaos theory

286

Minimum complexity of a chaotic system

Discrete chaotic systems, such as the
logistic map, can exhibit strange attractors
whatever their dimensionality. However, the
Poincare-Bendixson theorem shows that a
strange attractor can only arise in a
continuous dynamical system (specified by
differential equations) if it has three or more
dimensions. Finite dimensional linear
systems are never chaotic; for a dynamical
system to display chaotic behaviour it has to
be either nonlinear, or infinite-dimensional.

The Poincare-Bendixson theorem states that

a two dimensional differential equation has

very regular behavior. The Lorenz attractor

discussed above is generated by a system of

three differential equations with a total of

seven terms on the right hand side, five of

which are linear terms and two of which are quadratic (and therefore nonlinear). Another well-known chaotic

attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is

[31]
(quadratic) nonlinear. Sprott found a three dimensional system with just five terms on the right hand side, and

[32] [331

with just one quadratic nonlinearity, which exhibits chaos for certain parameter values. Zhang and Heidel
showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with
only three or four terms on the right hand side cannot exhibit chaotic behavior. The reason is, simply put, that
solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved.

While the Poincare-Bendixson theorem means that a continuous dynamical system on the Euclidean plane cannot be
chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behaviour. Perhaps

Bifurcation diagram of the logistic mapx — > r x (1 — x). Each vertical slice shows

the attractor for a specific value of r. The diagram displays period-doubling as r

increases, eventually producing chaos.

surprisingly, chaos may occur also in linear systems, provided they are infinite-dimensional
chaos is being developed in the functional analysis, a branch of mathematical analysis.

[34]

A theory of linear

History

Chaos theory

287

An early proponent of chaos theory was Henri Poincare. In the 1880s, while

studying the three-body problem, he found that there can be orbits which are

T351
nonperiodic, and yet not forever increasing nor approaching a fixed point.

In 1898 Jacques Hadamard published an influential study of the chaotic

motion of a free particle gliding frictionlessly on a surface of constant

T371
negative curvature. In the system studied, "Hadamard's billiards",

Hadamard was able to show that all trajectories are unstable in that all particle

trajectories diverge exponentially from one another, with a positive Lyapunov

exponent.

Much of the earlier theory was developed almost entirely by mathematicians,
under the name of ergodic theory. Later studies, also on the topic of nonlinear

Barnsley fern created using the chaos
game. Natural forms (ferns, clouds,
mountains, etc.) may be recreated
through an Iterated function system
(IFS).

T3R1

differential equations, were carried out by G.D. Birkhoff, A. N.
Kolmogorov, [39] [40] [41] M.L. Cartwright and J.E. Little wood, [42] and

[43]

Stephen Smale. Except for Smale, these studies were all directly inspired

by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of
Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had
not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio
circuits without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after
mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that
time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. What had
been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full
component of the studied systems.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of
chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by
hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to
visualize these systems.

An early pioneer of the theory was Edward Lorenz whose interest in

chaos came about accidentally through his work on weather prediction

T441
in 1961. Lorenz was using a simple digital computer, a Royal

McBee LGP-30, to run his weather simulation. He wanted to see a

sequence of data again and to save time he started the simulation in the

middle of its course. He was able to do this by entering a printout of

the data corresponding to conditions in the middle of his simulation

which he had calculated last time.

Turbulence in the tip vortex from an airplane
wing. Studies of the critical point beyond which a

system creates turbulence was important for
Chaos theory, analyzed for example by the Soviet

physicist Lev Landau who developed the
Landau-Hopf theory of turbulence. David Ruelle
and Floris Takens later predicted, against Landau,

that fluid turbulence could develop through a
strange attractor, a main concept of chaos theory.

To his surprise the weather that the machine began to predict was
completely different from the weather calculated before. Lorenz
tracked this down to the computer printout. The computer worked with
6-digit precision, but the printout rounded variables off to a 3-digit
number, so a value like 0.506127 was printed as 0.506. This difference
is tiny and the consensus at the time would have been that it should
have had practically no effect. However Lorenz had discovered that
small changes in initial conditions produced large changes in the

Chaos theory 288

[451
long-term outcome. Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed

atmospheric modelling cannot in general make long-term weather predictions. Weather is usually predictable only

about a week ahead.

The year before, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. Beforehand,
he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion

of noise-containing periods to error-free periods was a constant — thus errors were inevitable and must be planned for

T471
by incorporating redundancy. Mandelbrot described both the "Noah effect" (in which sudden discontinuous

changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly

[48] [49]

change afterwards). This challenged the idea that changes in price were normally distributed. In 1967, he

published "How long is the coast of Britain? Statistical self-similarity and fractional dimension", showing that a
coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in
length for an infinitesimally small measuring device. Arguing that a ball of twine appears to be a point when
viewed from far away (O-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand
(1 -dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An
object whose irregularity is constant over different scales ("self-similarity") is a fractal (for example, the Koch curve
or "snowflake", which is infinitely long yet encloses a finite space and has fractal dimension equal to circa 1.2619,
the Menger sponge and the Sierpihski gasket). In 1975 Mandelbrot published The Fractal Geometry of Nature,
which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial
systems proved to fit a fractal model.

Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol and in

[52] [53]

1958 by R.L. Ives. However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University,

Yoshisuke Ueda was experimenting with analog computers (that is, vacuum tubes) and noticed, on Nov. 27, 1961,
what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time,
and did not allow him to report his findings until 1970.

In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David
Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw (a physicist,
part of the Eudaemons group with J. Doyne Farmer and Norman Packard who tried to find a mathematical method to
beat roulette, and then created with them the Dynamical Systems Collective in Santa Cruz, California), and the
meteorologist Edward Lorenz.

The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of
Nonlinear Transformations", where he described logistic maps. Feigenbaum had applied fractal geometry to the
study of natural forms such as coastlines. Feigenbaum notably discovered the universality in chaos, permitting an
application of chaos theory to many different phenomena.

In 1979, Albert J. Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his
experimental observation of the bifurcation cascade that leads to chaos and turbulence in convective

Rayleigh— Benard systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum

T571
"for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".

Then in 1986 the New York Academy of Sciences co-organized with the National Institute of Mental Health and the

Office of Naval Research the first important conference on Chaos in biology and medicine. There, Bernardo

[50]
Huberman presented a mathematical model of the eye tracking disorder among schizophrenics. This led to a

renewal of physiology in the 1980s through the application of chaos theory, for example in the study of pathological

cardiac cycles.

[59]
In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters describing for

the first time self-organized criticality (SOC), considered to be one of the mechanisms by which complexity arises in

nature. Alongside largely lab-based approaches such as the Bak— Tang— Wiesenfeld sandpile, many other

investigations have focused on large-scale natural or social systems that are known (or suspected) to display

Chaos theory 289

scale-invariant behaviour. Although these approaches were not always welcomed (at least initially) by specialists in
the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of
natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of
scale-invariant behaviour such as the Gutenberg— Richter law describing the statistical distribution of earthquake
sizes, and the Omori law describing the frequency of aftershocks); solar flares; fluctuations in economic systems
such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires;
landslides; epidemics; and biological evolution (where SOC has been invoked, for example, as the dynamical
mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould).
Worryingly, given the implications of a scale-free distribution of event sizes, some researchers have suggested that
another phenomenon that should be considered an example of SOC is the occurrence of wars. These "applied"
investigations of SOC have included both attempts at modelling (either developing new models or adapting existing
ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or
characteristics of natural scaling laws.

The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced
the general principles of chaos theory as well as its history to the broad public. At first the domain of work of a few,
isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly
under the name of nonlinear systems analysis. Alluding to Thomas Kuhn's concept of a paradigm shift exposed in
The Structure of Scientific Revolutions (1962), many "chaologists" (as some described themselves) claimed that this
new theory was an example of such a shift, a thesis upheld by J. Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos
theory continues to be a very active area of research, involving many different disciplines (mathematics, topology,
physics, population biology, biology, meteorology, astrophysics, information theory, etc.).

Distinguishing random from chaotic data

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in
practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is
present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some
randomness.

All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system
always evolves in the same way from a given starting point. Thus, given a time series to test for determinism,

one can:

1. pick a test state;

2. search the time series for a similar or 'nearby' state; and

3. compare their respective time evolutions.

Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby
state. A deterministic system will have an error that either remains small (stable, regular solution) or increases
exponentially with time (chaos). A stochastic system will have a randomly distributed error.

Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test'
state (e.g., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on
phase space embedding methods. Typically one chooses an embedding dimension, and investigates the
propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you
can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it
is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more
computation time and a lot of data (the amount of data required increases exponentially with embedding dimension)
to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small
(less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the

Chaos theory 290

dimension too large — the method will work.

When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and
permanent distortions. Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new
dynamical properties. Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate
the deterministic part risk failure. Things become worse when the deterministic component is a non-linear feedback
system. In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear
series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.

The question of how to distinguish deterministic chaotic systems from stochastic systems has also been discussed in
philosophy.

Cultural references

Chaos theory has been mentioned in numerous movies and works of literature. Examples include the book Jurassic
Park, the film The Butterfly Effect, the sitcom The Big Bang Theory, Tom Stoppard's play Arcadia and the video
game Tom Clancy's Splinter Cell: Chaos Theory. The influence of chaos theory in shaping the popular understanding
of the world we live in was the subject of the BBC documentary High Anxieties - The Mathematics of Chaos directed
by David Malone. Chaos Theory is also the subject of discussion in the BBC documentary "The Secret Life of
Chaos" presented by the physicist Jim Al-Khalili.

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[2] Kellert, p. 56.
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Mathematica, vol. 13, pages 1—270.
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15-17(1991).
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Akademii Nauk SSSR 98: 527—530. See also Kolmogorov— Arnold— Moser theorem
[42] Mary L. Cartwright and John E. Littlewood (1945) "On non-linear differential equations of the second order, I: The equation y" + k(\—y )y'

+ y = bkkcos(kt + a), k large," Journal of the London Mathematical Society, vol. 20, pages 180—189. See also: Van der Pol oscillator
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43-49.
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[45] Gleick, James (1987). Chaos: Making a New Science. London: Cardinal, p. 17. ISBN 043429554X.
[46] Mandelbrot, Benoit (1963). "The variation of certain speculative prices". Journal of Business 36: pp. 394—419.
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Development, vol 7, pages 224—236.
[48] B. Mandelbrot, The Fractal Geometry of Nature (N.Y., N.Y.: Freeman, 1977), page 248.
[49] See also: Benoit B. Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward (N.Y.,

N.Y.: Basic Books, 2004), page 201.
[50] Benoit Mandelbrot (5 May 1967) "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension," Science,

Vol. 156, No. 3775, pages 636-638.
[51] B. van der Pol and J. van der Mark (1927) "Frequency demultiplication," Nature, vol. 120, pages 363—364. See also: Van der Pol oscillator
[52] R.L. Ives (10 October 1958) "Neon oscillator rings," Electronics, vol. 31, pages 108—115.
[53] See p. 83 of Lee W. Casperson, "Gas laser instabilities and their interpretation," pages 83—98 in: N. B. Abraham, F. T. Arecchi, and L. A.

Lugiato, eds., Instabilities and Chaos in Quantum Optics II: Proceedings of the NATO Advanced Study Institute, II Ciocco, Italy, June 28— July

7, 1987 (N.Y., N.Y.: Springer Verlag, 1988).

Chaos theory 292

[54] Ralph H. Abraham and Yoshisuke Ueda, eds., The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory (Singapore: World

Scientific Publishing Co., 2001). See Chapters 3 and 4.
[55] Sprott, J. Chaos and time-series analysis (http://books. google. com/books?id=SEDjdjPZ158C&pg=PA89&lpg=PA89&dq=ueda+

"Chihiro+Hayashi"&source=bl&ots=p9nZnB5MOD&sig=k2BrAnAyU84BHlMlrJ_-JC01mU0&hl=en&

ei=mYLrSr_MEI_KNcXcgIMM&sa=X&oi=book_result&ct=result&resnum=4&ved=0CA8Q6AEwAw#v=onepage&q=ueda "Chihiro

Hayashi"&f=false). Oxford. University Press, Oxford, UK, & New York, USA. 2003
[56] Mitchell Feigenbaum (July 1978) "Quantitative universality for a class of nonlinear transformations," Journal of Statistical Physics, vol. 19,

no. 1, pages 25—52.
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Scientific literature
Articles

• A.N. Sharkovskii, "Co-existence of cycles of a continuous mapping of the line into itself", Ukrainian Math. J.,
16:61-71 (1964)

• Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." American Mathematical Monthly 82, 985-92, 1975.

• Crutchfield, J. P., Farmer, J.D., Packard, N.H., & Shaw, R.S (December 1986). "Chaos". Scientific American 255
(6): 38—49 (bibliography p. 136) Online version (http://cse.ucdavis.edu/~chaos/courses/ncaso/Readings/
Chaos_SciAml986/Chaos_SciAml986.html) (Note: the volume and page citation cited for the online text differ
from that cited here. The citation here is from a photocopy, which is consistent with other citations found online,
but which don't provide article views. The online content is identical to the hardcopy text. Citation variations will
be related to country of publication).

• Kolyada, S. F. " Li- Yorke sensitivity and other concepts of chaos (http://www.springerlink.com/content/
q00627510552020g/?p=93elf3daf93549dl850365a8800afb30&pi=3)", Ukrainian Math. J. 56 (2004),
1242-1257.

Chaos theory 293

Textbooks

Alligood, K. T., Sauer, T., and Yorke, J. A. (1997). Chaos: an introduction to dynamical systems. Springer-Verlag

New York, LLC. ISBN 0-387-94677-2.

Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press.

ISBN 0-521-39511-9.

Badii, R.; Politi A. (1997). Complexity: hierarchical structures and scaling in physics (http://www. Cambridge.

org/catalogue/catalogue.asp?isbn=0521663857). Cambridge University Press. ISBN 0521663857.

Collet, Pierre, and Eckmann, Jean-Pierre (1980). Iterated Maps on the Interval as Dynamical Systems.

Birkhauser. ISBN 0-8176-4926-3.

Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd ed,. Westview Press.

ISBN 0-8133-4085-3.

Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0-521-47685-2.

Guckenheimer, J., and Holmes P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector

Fields. Springer-Verlag New York, LLC. ISBN 0-387-90819-6.

Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. Springer-Verlag New York, LLC.

ISBN 0-387-97173-4.

Hoover, William Graham (1999,2001). Time Reversibility, Computer Simulation, and Chaos. World Scientific.

ISBN 981-02-4073-2.

Kautz, Richard (2011). Chaos: The Science of Predictable Random Motion. Oxford University Press.

ISBN 978-0-19-959458-0.

Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing.

ISBN 0-472-08472-0.

Moon, Francis (1990). Chaotic and Fractal Dynamics. Springer-Verlag New York, LLC. ISBN 0-471-54571-6.

Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press New, York. ISBN 0-521-01084-5.

Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0-7382-0453-6.

Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9.

Tel, Tamas; Gruiz, Marion (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge

University Press. ISBN 0-521-83912-2.

Tufillaro, Abbott, Reilly (1992). An experimental approach to nonlinear dynamics and chaos. Addison-Wesley

New York. ISBN 0-201-55441-0.

Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press.

ISBN 0-198-52604-0.

Semitechnical and popular works

• Ralph H. Abraham and Yoshisuke Ueda (Ed.), The Chaos Avant-Garde: Memoirs of the Early Days of Chaos
Theory, World Scientific Publishing Company, 2001, 232 pp.

• Michael Barnsley, Fractals Everywhere, Academic Press 1988, 394 pp.

• Richard J Bird, Chaos and Life: Complexity and Order in Evolution and Thought, Columbia University Press
2003, 352 pp.

• John Briggs and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of
Wholeness, Harper Perennial 1990, 224 pp.

• John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper
Perennial 2000, 224 pp.

• Lawrence A. Cunningham, From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient
Capital Market Hypothesis, George Washington Law Review, Vol. 62, 1994, 546 pp.

• Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press
1988, 272 pp.

Chaos theory 294

James Gleick, Chaos: Making a New Science, New York: Penguin, 1988. 368 pp.

John Gribbin, Deep Simplicity,

L Douglas Kiel, Euel W Elliott (ed.), Chaos Theory in the Social Sciences: Foundations and Applications ,

University of Michigan Press, 1997, 360 pp.

Arvind Kumar, Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature , National

Book Trust, 2003.

Hans Lauwerier, Fractals, Princeton University Press, 1991.

Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996.

Chapter 5 of Alan Marshall (2002) The Unity of nature, Imperial College Press: London

Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp.

Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr

1991.

Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984.

Heinz-Otto Peitgen and P. H. Richter, The Beauty of Fractals : Images of Complex Dynamical Systems, Springer

1986,211pp.

David Ruelle, Chance and Chaos, Princeton University Press 1993.

Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993.

David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.

Peter Smith, Explaining Chaos, Cambridge University Press, 1998.

Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990.

Steven Strogatz, Sync: The emerging science of spontaneous order, Hyperion, 2003.

Yoshisuke Ueda, The Road To Chaos, Aerial Pr, 1993.

M. Mitchell Waldrop, Complexity : The Emerging Science at the Edge of Order and Chaos, Simon & Schuster,

1992.

External links

• Nonlinear Dynamics Research Group (http://lagrange.physics.drexel.edu) with Animations in Flash

• The Chaos group at the University of Maryland (http://www.chaos.umd.edu)

• The Chaos Hypertextbook (http://hypertextbook.com/chaos/). An introductory primer on chaos and fractals.

• Society for Chaos Theory in Psychology & Life Sciences (http://www.societyforchaostheory.org/)

• Nonlinear Dynamics Research Group at CSDC (http://www.csdc. unifi.it/mdswitch. html?newlang=eng),
Florence Italy

• Interactive live chaotic pendulum experiment (http://physics.mercer.edu/pendulum/), allows users to interact
and sample data from a real working damped driven chaotic pendulum

• Nonlinear dynamics: how science comprehends chaos (http://www.creatingtechnology.org/papers/chaos.
htm), talk presented by Sunny Auyang, 1998.

• Nonlinear Dynamics (http://www.egwald.ca/nonlineardynamics/index.php). Models of bifurcation and chaos
by Elmer G Wiens

• Gleicks Chaos (excerpt) (http://www.around.com/chaos.html)

• Systems Analysis, Modelling and Prediction Group (http://www.eng.ox.ac.uk/samp) at the University of
Oxford.

• A page about the Mackey-Glass equation (http://www.mgix.com/snippets/7MackeyGlass).

• High Anxieties - The Mathematics of Chaos (http://www.youtube.com/user/
thedebtgeneration?feature=mhum) (2008) BBC documentary directed by David Malone.

Chaos theory in organizational development 295

Chaos theory in organizational development

Chaos theory in organizational development refers to a subset of chaos theory which incorporates principles of
quantum mechanics and presents them in a complex systems environment.

Background

Viewing a chimp as a complex system in itself, and magnifying the interactional effects of primates and waves to
reflect the interactions of different elements making up a complex system, such as an organization, assists us in
seeing parallels between chaos theory and organizational relationships. What must be pointed out, however, is that
these "parallels" between organizations and the sub-atomic particles exist largely in terms of analogy
(metaphorically) between two very different domains of activity; the interactional effects of sub-atomic particles, in
quantum mechanics, are expressed in terms of math; bringing these theories into the domain of human activity can
be seen as problematical. Although these parallels are easily witnessed in regard to complex organizational systems,
it is difficult to see evidence of irrational quantum-effects in everyday life. If you roll a ball forward, it rolls forward
in the general direction intended. As a whole, Newtonian principles of interaction stand solidly within the bounds of
macrophysics. But at the sub-atomic level, things do not act as expected. "At the subatomic level, the objectivity
found in classical physics is replaced by quantum subjectivity." (Shelton, 2003) The introduction of chaos theory
brings the principles of quantum physics to the pragmatic world. These complex systems have a rather random
appearance and, until recently, have been labeled and discarded as chaotic and unintelligible. With the advent of
computer systems and powerful processors, it has become easier to map chaotic behavior and find interesting
underpinnings of order. The newly discovered underlying order to chaos sparked new interest and inspired more
research in the field of chaos theory. The recent focus of most of the research on chaos theory is primarily rooted in
these underlying patterns found in an otherwise chaotic environment, more specifically, concepts such as
self-organization, bifurcation, and self-similarity.

Elements of organization
Self-organization

Self-organization, as opposed to natural or social selection, is a dynamic change within the organization where
system changes are made by recalculating, re-inventing and modifying its structure in order to adapt, survive, grow,
and develop. Self-organization is the result of re-invention and creative adaptation due to the introduction of, or
being in a constant state of, perturbed equilibrium. One example of an organization which exists in a constant state of
perturbation is that of the learning organization, which is "one that allows self-organization, rather than attempting to
control the bifurcation through planned change." (Dooley, 1995) Being "off-balance" lends itself to regrouping and
re-evaluating the system's present state in order to make needed adjustments and regain control and equilibrium. By
understanding and introducing the element of punctuated equilibrium (chaos) while facilitating networks for growth,
an organization can change gears from "cruise" to "turbo" in regard to speed and intensity of organizational change.
While maintaining an equilibrial state seems to be an intuitively rational method for enabling an organization to gain
a sense of consistency and solidarity, existing on the edge of a chaotic state remains the most beneficial environment
for systems to flourish develop and grow.

For instance, two competing organizations that differ in regard to their levels of homeostasis will not be in
competition for long. Generally speaking, the organization with the less-stable structure will come out ahead while
the constant stability of the latter will eventually lead to its own demise. Although quite similar, small differences in
homeostasis levels are enough to make a tremendous difference in future outcomes for each organization. The notion
of similarity in origin vs. dissimilar results comes to fruition with the emergence of bifurcation.

Chaos theory in organizational development 296

Bifurcation

The concept of bifurcation cannot be explained without discussion of the term frequently labeled "sensitivity to
initial conditions." Sensitivity to initial conditions refers to the high level of importance of primary conditions from
which the future path and direction of a system stems. This sensitivity to initial conditions is commonly referred to
as the "Butterfly Effect," in which a butterfly flaps its tiny wings in one end of the world which results in a typhoon
or hurricane somewhere else on the globe. While this is an entertaining notion, sensitivity to initial conditions
remains in reality a very abstract concept without the presence of bifurcation, which is mathematically labeled as the
actual splitting point of two near-identical entities which, due to the sensitivity of initial conditions, tend to take two
very distinct paths and result in two totally different geographically or even evolutionary places.

Imagine dropping two identical coins from your fingertips off a 25-story balcony at the same time. Unless they are
glued together, they will each take a different path towards the ground. Even though the force of gravity determines
their general direction and speed, a host of uncontrollable variables such as wind and dust particles affect each coin
independently. The infinitesimal and perhaps unidentifiable difference in starting conditions exponentially amplifies
the effects of all other variables encountered which then feed back and add even more variation to the system
resulting in very different paths taken to the ground. The moment the two coins split paths is known as the
bifurcation point. The importance of this point lies in its implication of change and new direction.

Applications and pitfalls

The primary goal of an organizational development (OD) consultant is to initiate, facilitate, and support successful
change in an organization. Using chaos theory as the sole model for change may be far too risky for any stakeholder
buy-in. The concept of uncertainty on which chaos theory relies is not an appealing motive for change compared to
many alternative "safer" models of organizational change which entail less risk. By careful planning and
management of disorder a successful intervention is possible, but only with a truly dedicated arsenal of talented and
creative resources. By permitting or actively forcing an organization to enter a chaotic state, change becomes
inevitable and bifurcation imminent; but the question remains, "Will the new direction be the one intended?" In order
to account for the direction of the new thrust, most planning attention should be focused on attractors instead of the
initiation of disorder.

Although chaos eventually gives way to self-organization, how can we control the duration, intensity, and shape of
its outcome? It seems that punctuating equilibrium and instilling disorder in an organization is risky business.
Throwing an organization off balance could possibly send it in a downward spiral towards dissemination by
ultimately compromising the structural integrity (i.e. identity) of the system to the point of no return. The only way
to reap the benefits of chaos theory in OD while maintaining a sense of security is to adjust the organization towards
a state of existence which lies "on the edge of chaos".

By existing on the edge of chaos, organizations are forced to find new, creative ways to compete and stay ahead.
Good examples of such learning organizations are found throughout the field of technology as well as the airline
industry, namely organizations such as Southwest Airlines, which used re-invention not just for survival, but also to
prosper in an otherwise dismal market. In contrast, there are organizations which, due to extended periods of
equilibrium, find themselves struggling for survival. Telephone companies, for instance, were once solid and static
entities that dominated the communication market. While the rest of the world was developing new communication
technology, telephone companies did not creatively grow at the same rate. The result is an organization that is
battling to stay alive unless they embrace the element of chaos due to crisis, and allow creative adaptability to
function freely so that self-organization and re-invention can occur.

While organizations existing on the edge of chaos are known to be the most creative and adaptive of organizations,
how do their members feel about constant evolution and re-invention? Is it possible to identify with, and stay loyal
to, an organization that constantly changes shape? The short answer is yes. As long as the organization does not
change its core essence, its identifiable, shared purpose, its members will still experience the organization as a

Chaos theory in organizational development 297

developing system that changes shape but retains the same familiar face.

Perhaps the safest way to use chaos theory in OD is not in the instigation of organizational change, but in the use of
its principles in dealing with issues that arise within the organization. By embracing organizational phenomena
previously seen as dysfunctional, such as interpersonal conflict, and using it as a source for transformational change
by applying principles found in chaos theory (Shelton, 2003), an organization can make "lemonade out of lemons"
and become more responsive to change agents while continuously moving ahead and growing from the inside out
without the fear of complete chaos.

References

Cornejo Alvarez, Alfonso (1997). Caos y Complejidad. (Free online book. Translated. Originally in Spanish.

DeShon, R. & Svyantek, D. J. (1993) "Organizational Attractors: A Chaos Theory Explanation of why Cultural

Change Efforts Often Fail." Public Administration Quarterly. (Vol. 17, No. 3 pp. 339-355)

Dooley, K. & Johnson, L. (1995) "TQM, Chaos, and Complexity." Human Systems Management. (Vol. 14, No. 4,

pp. 1-16)

Higgins, M. & Smith, W. (2003). "Postmodernism and Popularisation: The Cultural Life of Chaos Theory."

Culture and Organization. (Vol. 9, June, pp. 93-104). Leicester, UK: Management Centre.

Hudson, C.G. (2000). "At the Edge of Chaos: A New Paradigm for Social Work?" Journal of Social Work

Education (Vol. 36, No. 2, pp. 215-230)

Levy, D. (1994). "Chaos Theory and Strategy: Theory, Application, and Managerial Implications." Strategic

Management Journal. (Vol. 15, pp. 167-178).

T21
Mc Namara, C.R. (1997). "Brief Overviews of Contemporary Theories in Management." Retrieved March 15,

2005.

Polley, D. (1997) "Turbulence in Organizations: New Metaphors for Organizational Research." Organization

Science. (Vol. 8, No.5, pp. 445-457)

Shaw, P. (1997). "Intervening in the Shadow Systems of Organizations." Journal of Organizational Change

Management. (Vol. 10, No. 3, pp. 235-250). Hertfordshire, UK: MCB University Press

Shelton, C. (2003). "From Chaos to Order: Exploring New Frontiers in Conflict Management." (Presented at the

Midwest Academy of Management Conference). Kansas City: Charlotte Shelton & John Darling

Shermer, M. (1995). "Exorcising Laplace's Demon: Chaos and Antichaos, History and Metahistory." History and

Theory, (Vol. 34, No. 1 pp. 59-83). Oxford, UK: Blackwell Publishing

Smith, W. (2001). "Chaos Theory and Postmodern Organization." International Journal of Organizational Theory

and Behavior. (Vol. 4, pp. 159—286)

Wheatley, Margaret J. (2006). Leadership and the New Science: Discovering Order in a Chaotic World. (3rd Ed.)

Berrett-Koehler Publishers, Inc.

References

[1] http://translate.google.com/translate?hl=en&sl=es&u=http://www.eumed.net/cursecon/libreria/2004/aca/aca.htm&sa=X&

oi=translate&resnum=2&ct=result&prev=/

search%3Fq%3Dcaos%2By%2Bcomplejidad%2Bmonterrey%26hl%3Den%26client%3Dsafari%26rls%3Den-us%26sa%3DG
[2] http : // w w w . managementhelp . org/mgmnt/cntmpory . htm

Attractor

298

Attractor

An attractor is a set towards which a
dynamical system evolves over time.
That is, points that get close enough to
the attractor remain close even if
slightly disturbed. Geometrically, an
attractor can be a point, a curve, a
manifold, or even a complicated set
with a fractal structure known as a
strange attractor. Describing the
attractors of chaotic dynamical systems
has been one of the achievements of
chaos theory.

A trajectory of the dynamical system

in the attractor does not have to satisfy

any special constraints except for

remaining on the attractor. The

trajectory may be periodic or chaotic

or of any other type. If this condition is met, but the flow in the neighbourhood is away from the set, the set is called

a repeller (or repellor).

Visual representation of a strange attractor

Motivation

A dynamical system is generally described by one or more differential or difference equations. The equations of a
given dynamic system specify its behavior over any given short period of time. To determine the system's behavior
for a longer period, it is necessary to integrate the equations, either through analytical means or through iteration,
often with the aid of computers.

Dynamical systems in the physical world tend to be dissipative: if it were not for some driving force, the motion
would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many
causes.) The dissipation and the driving force tend to combine to kill out initial transients and settle the system into
its typical behavior. This one part of the phase space of the dynamical system corresponding to the typical behavior
is the attracting section or attractee.

Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under
the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial
state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are
limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set,
but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of
the limit set.

For example, the damped pendulum has two invariant points: the point Xrjof minimum height and the point X\oi
maximum height. The point Xois also a limit set, as trajectories converge to it; the point X\is not a limit set.
Because of the dissipation, the point Xrjis also an attractor. If there were no dissipation, Xrjwould not be an
attractor.

Attractor 299

Mathematical definition

Let/(f, •) be a function which specifies the dynamics of the system. That is, if a is a point in the phase space, so that
the state of the system at a certain time, then f(0, a) = a and for a positive value of t, f(t, a) is the result of the
evolution of this state after t units of time. For example, if the system is a free particle in one dimension then the

phase space is the plane R with coordinates (x,v), where x is the position of the particle and v is its velocity, and the
evolution is given by

f(t,(x,v)) = (x + tv,v).
An attractor is a subset A of the phase space characterized by the following three conditions:

• A is forward invariant under/: if a is an element of A then so isf{t,a), for all t > 0.

• There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all
points b that "enter A in the limit t — > °°". More formally, B(A) is the set of all points b in the phase space with the
following property:

For any open neighborhood N of A, there is a positive constant T such that f(t,b) £ N for all real t > T.

• There is no proper subset of A having the first two properties.

Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted
to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on
the topology of the phase space. In the case of R", the Euclidean norm is typically used.

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have
positive measure (preventing a point from being an attractor), others relax the requirement that B(A) be a
neighborhood.

Types of attractors

Attractors are parts of the phase space of the dynamical system. Until the 1960s, as evidenced by textbooks of that
era, attractors were thought of as being geometrical subsets of the phase space: points, lines, surfaces, volumes. The
(topologically) wild sets that had been observed were thought to be fragile anomalies. Stephen Smale was able to
show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Two simple attractors are the fixed point and the limit cycle. There can be many other geometrical sets that are
attractors. When these sets (or the motions on them), are hard to describe, then the attractor is a strange attractor, as
described in the section below.

Fixed point

A fixed point is a point of a function that does not change under some

transformation. If we regard the evolution of a dynamical system as a

series of transformations, then there may or may not be a point which

remains fixed under the whole series of transformation. In general

there would not be such a point, but there may be one. The final state

that a dynamical system evolves towards, such as the final states of a

falling pebble, a damped pendulum, or the water in a glass corresponds Weakly attracting fixed point for complex

to a fixed point of the evolution function, and will occur at the quadratic polynomial

attractor, but the two concepts are not equivalent. A marble rolling

around in a basin may have a fixed point in phase space even if it doesn't in physical space. Once it has lost

momentum and settled into the bottom of the bowl it then has a fixed point in physical space, phase space, and is

located at the attractor for that system.

Attractor

300

Limit cycle

See main article limit cycle

A limit cycle is a periodic orbit of the system that is isolated. Examples include the swings of a pendulum clock, the
tuning circuit of a radio, and the heartbeat while resting. The ideal pendulum is not an example because its orbits are
not isolated. In phase space of the ideal pendulum, near any point of a periodic orbit there is another point that
belongs to a different periodic orbit.

-2-1012

Van der Pol oscillatorVan der Pol phase portrait

Limit tori

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. If
two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed,
and the limit cycle becomes a limit torus. We call this kind of attractor JV^ -torus if there are JVj incommensurate
frequencies. For example here is a 2-torus:

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of JVj periodic
functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict
periodicity, but its power spectrum still consists only of sharp lines.

Attractor

301

Strange attractor

An attractor is informally described as strange if it has
non-integer dimension. This is often the case when the dynamics
on it are chaotic, but there exist also strange attractors that are not
chaotic. The term was coined by David Ruelle and Floris Takens
to describe the attractor that resulted from a series of bifurcations
of a system describing fluid flow. Strange attractors are often
differentiable in a few directions, but some are like a Cantor dust,
and therefore not differentiable.

Examples of strange attractors include the Henon attractor, Rossler
attractor, Lorenz attractor, Tamari attractor.

Partial differential equations

Parabolic partial differential equations may have

finite-dimensional attractors. The diffusive part of the equation

damps higher frequencies and in some cases leads to a global attractor. The Ginzburg— Landau, the

Kuramoto—Sivashinsky, and the two-dimensional, forced Navier— Stokes equations are all known to have global

attractors of finite dimension.

For the three-dimensional, incompressible Navier— Stokes equation with periodic boundary conditions, if it has a
global attractor, then this attractor will be of finite dimensions.

A plot of Lorenz's strange attractor for values p=28, o
= 10, |3 = 8/3

References

Attractor at Scholarpedia, curated by John Milnor.

David Ruelle and Floris Takens (1971). "On the nature of turbulence". Communications of Mathematical Physics

20: 167-192. doi:10.1007/BF01646553.

D. Ruelle (1981). "Small random perturbations of dynamical systems and the definition of attractors".

Communications of Mathematical Physics 82: 137-151. doi:10.1007/BF01206949.

John Milnor (1985). "On the concept of attractor". Communications of Mathematical Physics 99: 177—195.

doi:10.1007/BF01212280.

David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press.

ISBN 0-12-601710-7.

Ruelle, David (August 2006). "What is. ..a Strange Attractor?" (PDF). Notices of the American Mathematical

Society 53 (7): 764-765. Retrieved 2008-01-16.

Ben Tamari (1997). Conservation and Symmetry Laws and Stabilization Programs in Economics. Ecometry ltd.

ISBN 965-222-838-9.

Grebogi, Ott, Pelikan, Yorke (1984). "Strange attractors that are not chaotic". Physica D 13: 261—268.

doi: 10. 1016/0167-2789(84)90282-3.

Attractor 302

Further reading

• Edward N. Lorenz (1996) The Essence of Chaos ISBN 0-295-97514-8

• James Gleick (1988) Chaos: Making a New Science ISBN 0-140-09250-1

External links

[31

• Basin of attraction on Scholarpedia

• A gallery of trigonometric strange attractors

• A gallery of polynomial strange attractors

• Animated Pickover Strange Attractors

[71

• Chaoscope, a 3D Strange Attractor rendering freeware

ro]

• ID, 2D and 3D of strange attractors, include Tamari Attractor

• Research abstract and software laboratory

• A java generator for strange attractors

ri2i

• Online strange attractors generator

• Tamari attractor

References

[I] http://www.scholarpedia.org/article/Attractor

[2] http://www.ams.org/notices/200607/what-is-ruelle.pdf

[3] http://www.scholarpedia.org/article/Basin_of_attraction

[4] http://slide.nethium.pl/album_en.net7gNwADMfFmY

[5] http://ccrma-www.stanford.edu/~stilti/images/chaotic_attractors/poly.html

[6] http://www.aidansamuel.com/strange.php

[7] http://www.chaoscope.org

[8] http://www.ecometry.biz/attractors

[9] http://ronrecord.com/PhD/intro.html

[10] ftp://ftp2.sco.com/pub/skunkware/src/xll/misc/mathrec-l.lc.tar.gz

[II] http://www.dse.nl/~rolandb/attractort/attractor.html
[12] http://wokos.nethium.pl/attractors_en.net

[13] http://www.bentamari.com/attractors.html

Lorenz attractor

303

Lorenz attractor

The Lorenz attractor, named for Edward N. Lorenz, is an
example of a non-linear dynamic system corresponding to the
long-term behavior of the Lorenz oscillator. The Lorenz oscillator
is a 3-dimensional dynamical system that exhibits chaotic flow,
noted for its lemniscate shape. The map shows how the state of a
dynamical system (the three variables of a three-dimensional
system) evolves over time in a complex, non-repeating pattern.

A plot of the trajectory Lorenz system for values p=28

10, (3 = 8/3

Overview

The attractor itself, and the equations from which it is derived,
were introduced in 1963 by Edward Lorenz, who derived it from
the simplified equations of convection rolls arising in the
equations of the atmosphere.

In addition to its interest to the field of non-linear mathematics, the
Lorenz model has important implications for climate and weather
prediction. The model is an explicit statement that planetary and
stellar atmospheres may exhibit a variety of quasi-periodic
regimes that are, although fully deterministic, subject to abrupt and
seemingly random change.

From a technical standpoint, the Lorenz oscillator is nonlinear,

three-dimensional and deterministic. For a certain set of

parameters, the system exhibits chaotic behavior and displays what is today called a strange attractor. The strange

attractor in this case is a fractal of Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the

Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01.

A trajectory of Lorenz's equations, rendered as a metal
wire to show direction and 3D structure

The system also arises in simplified models for lasers (Haken 1975) and dynamos (Knobloch 1981).

Lorenz attractor

304

Equations

The equations that govern the Lorenz oscillator are:

Trajectory with scales added

~di
dy

dt
dz

~dl

a(y - x)
x{p -z)-y

= xy — 3z

where <ris called the Prandtl number and pis called the Rayleigh number. All u , p, 8 > 0, but usually
U = 10, 3 = 8/3and pis varied. The system exhibits chaotic behavior for p = 28but displays knotted

periodic orbits for other values of p. For example, with p = 99.96it becomes a 7(3,2) torus knot.

When o*0 and |3 (p-1) t 0, the equations generate three critical points. The critical points at (0,0,0) correspond to

no convection, and the critical points at (±*//3(p — 1) ±i//?(o — 1) p — l)correspond to steady convection.

<t + /5 + 3

This pair is stable only if p

(3-1

, which can hold only for positive pif a > 8 -\- 1.

Sensitive dependence on the initial condition

Time t=l (Enlarge)

Time t=2 (Enlarge)

Time t=3 (Enlarge)

These figures — made using p=28, o = 10 and |3 = 8/3 — show three time segments of the 3-D evolution of 2 trajectories (one in blue, the

other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10~~ in the x-coordinate. Initially, the two trajectories

seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.

Java animation of the Lorenz attractor shows the continuous evolution.

[1]

Lorenz attractor

305

Rayleigh number

The Lorenz attractor for different values of p

p=14, a=10, p=8/3 (Enlarge)

p=13, o=10, p=8/3 (Enlarge)

p=15, o=10, p=8/3 (Enlarge)

p=28, o=10, p=8/3 (Enlarge)

For small values of p, the system is stable and evolves to one of two fixed point attractors. When p is larger than 24.28, the fixed
points become repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself.

Java animation showing evolution for different values of p

[1]

Source code

The source code to simulate the Lorenz attractor in GNU Octave follows.

% Lorenz Attractor equations solved by

0DE Solve

%% x ' = sigma* (y-x)

%% y ' = x* (rho - z) - y

%% z ' = x*y - beta*z

function dx = lorenzatt (X)

rho = 28; sigma = 10; beta = 8/3;

dx = zeros (3,1);

dx (1) = sigma* (X (2)

- X(l));

dx(2) = X(l) * (rho -

X(3)) - X(2);

dx(3) = X(l) *X(2)

beta*X(3) ;

return

end

% Using LS0DE to solve the ODE system.

clear all

close all

lsode_options ("absolute

tolerance" , le-

Lorenz attractor 306

lsode_options ("relative

tol

erance'

,le-4)

t = linspace

(0,25, le3) ;

[0,1,

1.05];

[X,T,MSG]=lsode (@lorenzatt,

,t) ;

MSG

plot3 (X(: , 1)

,X(:,2),X(:

,3))

view(45, 45)

References

Jonas Bergman, Knots in the Lorentz Equation , Undergraduate thesis, Uppsala University 2004.

Fr0yland, J., Alfsen, K. H. (1984). "Lyapunov-exponent spectra for the Lorenz model". Phys. Rev. A 29 (5):

2928-293 1 . doi: 10. 1 103/PhysRevA.29.2928.

P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica D 9 (1-2):

189-208. Bibcode 1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-l.

Haken, H. (1975). "Analogy between higher instabilities in fluids and lasers". Physics Letters A 53 (1): 77—78.

doi: 10. 1016/0375-9601(75)90353-9.

Lorenz, E. N. (1963). "Deterministic nonperiodic flow". J. Atmos. Sci. 20 (2): 130—141.

doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

Knobloch, Edgar (1981). "Chaos in the segmented disc dynamo". Physics Letters A 82 (9): 439—440.

doi:10.1016/0375-9601(81)90274-7.

Strogatz, Steven H. (1994). Nonlinear Systems and Chaos. Perseus publishing.

Tucker, W. (2002). "A Rigorous ODE Solver and Smale's 14th Problem" [3] . Found. Comp. Math. 2: 53-117.

External links

mi
Weisstein, Eric W., "Lorenz attractor from Math World.

Lorenz attractor by Rob Morris, Wolfram Demonstrations Project.

Lorenz equation on planetmath.org

For drawing the Lorenz attractor, or coping with a similar situation using ANSI C and gnuplot.

ro]

Synchronized Chaos and Private Communications, with Kevin Cuomo . The implementation of Lorenz

attractor in an electronic circuit

nation (y

[10]

[91
Lorenz attractor interactive animation (you need the Adobe Shockwave plugin)

Levitated.net: computational art and design

3D Attractors: Mac program to visualize and explore the Lorenz attractor in 3 dimensions

[121

3D VRML Lorenz attractor (you need a VRML viewer plugin)

[131

Essay on Lorenz attractors in J - see J programming language

[141

Applet for non-linear simulations (select "Lorenz attractor" preset), written by Viktor Bachraty in Jython
Lorenz Attractor implemented in analog electronic

Visualizing the Lorenz attractor in 3D with Python and VTK

[171
Lorenz Attractor implemented in Flash

Lorenz attractor

307

References

[I] http://to-campos.planetaclix.pt/fractal/lorenz_eng.html
[2] http://www.teorfys.uu.se/en/node/467

[3] http://www.math.uu.se/~warwick/main/rodes.html

[4] http://mathworld.wolfram.com/LorenzAttractor.html

[5] http://demonstrations.wolfram.com/LorenzAttractor/

[6] http://planetmath.org/encyclopedia/LorenzEquation.html

[7] http://www.mizuno.Org/c/la/index.shtml

[8] http://video.google.com/videoplay?docid=2875296564158834562&q=strogatz&ei=xr90SJ_SOpeG2wKB3Iy2DA&hl=en

[9] http://toxi.co.uk/lorenz/

[10] http ://www. levitated. net/daily/levLorenzAttractor. html

[II] http://amath.colorado.edu/faculty/juanga/3DAttractors.html
[12] http://ibiblio.org/e-notes/VRML/Lorenz/Lorenz.htm

[13] http://www.jsoftware.com/jwiki/Essays/Lorenz_Attractor

[ 14] http ://student. fiit . stuba. sk/~bachratv02/mes/applet.html

[15] http://frank.harvard.edu/~paulh/misc/lorenz.htm

[16] http://www.martinlaprise.info/2010/02/28/visualizing-the-lorentz-attractor-with-vtk/

[17] http://911web.org/lorenz-attractor.php

Strange attractor

A trajectory of the dynamical system

in the attractor does not have to satisfy

any special constraints except for

remaining on the attractor. The

trajectory may be periodic or chaotic

or of any other type. If this condition is met, but the flow in the neighbourhood is away from the set, the set is called

a repeller (or repellor).

Visual representation of a strange attractor

Strange attractor 308

Motivation

For example, the damped pendulum has two invariant points: the point Xoof minimum height and the point X\oi
maximum height. The point Xois also a limit set, as trajectories converge to it; the point X\\s not a limit set.
Because of the dissipation, the point Xois also an attractor. If there were no dissipation, Xowould not be an
attractor.

Mathematical definition

Let/(f, •) be a function which specifies the dynamics of the system. That is, if a is a point in the phase space, so that
the state of the system at a certain time, then f(0, a) = a and for a positive value of t, f{t, a) is the result of the
evolution of this state after t units of time. For example, if the system is a free particle in one dimension then the

phase space is the plane R with coordinates (x,v), where x is the position of the particle and v is its velocity, and the
evolution is given by

f(t,(x,v)) = (x + tv,v).
An attractor is a subset A of the phase space characterized by the following three conditions:

• A is forward invariant under/: if a is an element of A then so isf(t,a), for all t > 0.

• There exists a neighborhood of A, called the basin of attraction for A and denoted B{A), which consists of all
points b that "enter A in the limit t — > °°". More formally, B{A) is the set of all points b in the phase space with the
following property:

For any open neighborhood N of A, there is a positive constant T such that f(t,b) € N for all real t > T.

• There is no proper subset of A having the first two properties.

Strange attractor

309

Types of attractors

Fixed point

A fixed point is a point of a function that does not change under some

transformation. If we regard the evolution of a dynamical system as a

series of transformations, then there may or may not be a point which

remains fixed under the whole series of transformation. In general

there would not be such a point, but there may be one. The final state

that a dynamical system evolves towards, such as the final states of a

falling pebble, a damped pendulum, or the water in a glass corresponds

to a fixed point of the evolution function, and will occur at the

attractor, but the two concepts are not equivalent. A marble rolling

around in a basin may have a fixed point in phase space even if it doesn't in physical space. Once it has lost

momentum and settled into the bottom of the bowl it then has a fixed point in physical space, phase space, and is

located at the attractor for that system.

Weakly attracting fixed point for complex
quadratic polynomial

Limit cycle

See main article limit cycle

-2-1012

Van der Pol oscillatorVan der Pol phase portrait

Strange attractor

310

Limit tori

Strange attractor

Examples of strange attractors include the Henon attractor, Rossler
attractor, Lorenz attractor, Tamari attractor.

Partial differential equations

A plot of Lorenz's strange attractor for values p=28, o
= 10, p = 8/3

Parabolic partial differential equations may have

finite-dimensional attractors. The diffusive part of the equation

damps higher frequencies and in some cases leads to a global attractor. The Ginzburg— Landau, the

Kuramoto—Sivashinsky, and the two-dimensional, forced Navier— Stokes equations are all known to have global

attractors of finite dimension.

For the three-dimensional, incompressible Navier— Stokes equation with periodic boundary conditions, if it has a
global attractor, then this attractor will be of finite dimensions.

Strange attractor 311

References

Attractor at Scholarpedia, curated by John Milnor.

David Ruelle and Floris Takens (1971). "On the nature of turbulence". Communications of Mathematical Physics

20: 167-192. doi:10.1007/BF01646553.

D. Ruelle (1981). "Small random perturbations of dynamical systems and the definition of attractors".

Communications of Mathematical Physics 82: 137-151. doi:10.1007/BF01206949.

John Milnor (1985). "On the concept of attractor". Communications of Mathematical Physics 99: 177—195.

doi:10.1007/BF01212280.

David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press.

ISBN 0-12-601710-7.

Ruelle, David (August 2006). "What is. ..a Strange Attractor?" (PDF). Notices of the American Mathematical

Society 53 (7): 764-765. Retrieved 2008-01-16.

Ben Tamari (1997). Conservation and Symmetry Laws and Stabilization Programs in Economics. Ecometry ltd.

ISBN 965-222-838-9.

Grebogi, Ott, Pelikan, Yorke (1984). "Strange attractors that are not chaotic". Physica D 13: 261—268.

doi: 10. 1016/0167-2789(84)90282-3.

Further reading

• Edward N. Lorenz (1996) The Essence of Chaos ISBN 0-295-97514-8

• James Gleick (1988) Chaos: Making a New Science ISBN 0-140-09250-1

External links

T31
Basin of attraction on Scholarpedia

A gallery of trigonometric strange attractors

A gallery of polynomial strange attractors

Animated Pickover Strange Attractors

Chaoscope, a 3D Strange Attractor rendering freeware

ID, 2D and 3D of strange attractors, include Tamari Attractor

Research abstract and software laboratory

A java generator for strange attractors

Online strange attractors generator

Tamari attractor

Butterfly effect

312

Butterfly effect

In chaos theory, the butterfly effect is the

sensitive dependence on initial conditions;
where a small change at one place in a
nonlinear system can result in large
differences to a later state. For example, the
presence or absence of a butterfly flapping
its wings could lead to creation or absence
of a hurricane.

Although the butterfly effect may appear to
be an esoteric and unusual behavior, it is
exhibited by very simple systems: for
example, a ball placed at the crest of a hill
might roll into any of several valleys
depending on slight differences in initial
position.

The butterfly effect is a common trope in
fiction when presenting scenarios involving
time travel and with "what if" cases where
one storyline diverges at the moment of a

Sensitive dependency
on initial conditions

jf A v attractor D

Key: Blue squares represent initial states;
black circles represent equilibria

Point attractors in 2D phase space.

seemingly minor event resulting in two significantly different outcomes.

Theory

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on
initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making
complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case
of weather), since it is impossible to measure the starting atmospheric conditions completely accurately.

Origin of the concept and the term

The term "butterfly effect" itself is related to the work of Edward Lorenz, and it is based in chaos theory and
sensitive dependence on initial conditions, already described in the literature in a particular case of the three-body
problem by Henri Poincare in 1890. He later proposed that such phenomena could be common, say in
meteorology. In 1898, Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature,
and Pierre Duhem discussed the possible general significance of this in 1908. The idea that one butterfly could
eventually have a far-reaching ripple effect on subsequent historic events seems first to have appeared in "A Sound
of Thunder", a 1952 short story by Ray Bradbury about time travel (see Literature and print here) although Lorenz
made the term popular. In 1961, Lorenz was using a numerical computer model to rerun a weather prediction, when,
as a shortcut on a number in the sequence, he entered the decimal .506 instead of entering the full .506127. The
result was a completely different weather scenario. Lorenz published his findings in a 1963 paper for the New
York Academy of Sciences noting that "One meteorologist remarked that if the theory were correct, one flap of a
seagull's wings could change the course of weather forever." Later speeches and papers by Lorenz used the more
poetic butterfly. According to Lorenz, when Lorenz failed to provide a title for a talk he was to present at the 139th
meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the

Butterfly effect

313

flap of a butterfly's wings in Brazil set off a tornado in Texas? as a title. Although a butterfly flapping its wings has
remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location

of the consequences have varied widely.

The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately
alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in another location. The
flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading
to large-scale alterations of events (compare: domino effect). Had the butterfly not flapped its wings, the trajectory of
the system might have been vastly different. While the butterfly does not "cause" the tornado in the sense of
providing the energy for the tornado, it does "cause" it in the sense that the flap of its wings is an essential part of the
initial conditions resulting in a tornado, and without that flap that particular tornado would not have existed.

Illustration

The butterfly effect in the Lorenz attractor

time < t < 30 (larger)

Z coordinate (larger)

These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of
time in the Lorenz attractor starting at two initial points that differ only by 10 in the x-coordinate. Initially, the two trajectories seem coincident, as
indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of
the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at f=30.

A Java animation of the Lorenz attractor shows the continuous evolution.

Mathematical definition

A dynamical system with evolution map f * displays sensitive dependence on initial conditions if points arbitrarily

close together become separate with increasing t at an exponential rate. The definition is not topological, but
essentially metrical.

If M is the state space for the map f t , then f * displays sensitive dependence to initial conditions if for any x in M
and any 6>0, there are y in M, with < d(x, y) < 5 such that

d{f T {x)J T {y)) > exp(ar)d(x,y).
The definition does not require that all points from a neighborhood separate from the base point x, but it requires one
positive Lyapunov exponent.

Butterfly effect 314

Examples

The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction
models, for example.

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of
cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem.
Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure
quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is

included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers.

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the

time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of

ri2i
sensitivity of quantum systems to small changes in their given Hamiltonians. Poulin et al. present a quantum

algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected

to slightly different dynamics." They consider fidelity decay to be "the closest quantum analog to the (purely

ri3i

classical) butterfly effect. Whereas the classical butterfly effect considers the effect of a small change in the
position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect
of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly

effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to
initial conditions are known as quantum chaos.

References

[I] Some Historical Notes: History of Chaos Theory (http://www.wolframscience.com/reference/notes/971c)
[2] Mathis, Nancy (2007). Storm Warning: The Story of a Killer Tornado. Touchstone, p. x. ISBN 0743280532.

[3] Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow" (http://journals.ametsoc.org/doi/abs/10.1175/

1520-0469(1963)020<0130:DNF>2.0.CO;2). Journal of the Atmospheric Sciences 20 (2): 130-141.

doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. . Retrieved 3 June 2010.
[4] "Butterfly Effects - Variations on a Meme" (http://clearnightsky.com/node/428). clearnightsky.com (http://www.clearnightsky.com). .
[5] http://www.realclimate.org/index.php/archives/2005/ll/chaos-and-climate/
[6] Heller, E. J.; Tomsovic, S. (July 1993). "Postmodern Quantum Mechanics". Physics Today.

[7] Gutzwiller, Martin C. (1990). Chaos in Classical and Quantum Mechanics. New York: Springer-Verlag. ISBN 0387971734.
[8] Rudnick, Ze'ev (January 2008). "What is... Quantum Chaos" (http://www.ams.org/notices/200801/tx080100032p.pdf) (PDF). Notices of

the American Mathematical Society. .
[9] Berry, Michael (1989). "Quantum chaology, not quantum chaos". Physica Scripta 40: 335. doi: 10. 1088/003 1-8949/40/3/013.
[10] Gutzwiller, Martin C. (1971). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical Physics 12: 343.

doi: 10.1063/1.1665596.

[II] Gao, J.; Delos, J. B. (1992). "Closed-orbit theory of oscillations in atomic photoabsorption cross sections in a strong electric field. II.
Derivation of formulas". Phys. Rev. A 46 (3): 1455-1467. doi: 10.1 103/PhysRevA.46.1455.

[12] Karkuszewski, Zbyszek P.; Jarzynski, Christopher; Zurek, Wojciech H. (2002). "Quantum Chaotic Environments, the Butterfly Effect, and

Decoherence". Physical Review Letters 89 (17): 170405. Bibcode 2002PhRvL..89q0405K. doi:10.1 103/PhysRevLett.89.170405.
[13] Poulin, David; Blume-Kohout, Robin; Laflamme, Raymond; Ollivier, Harold (2004). "Exponential Speedup with a Single Bit of Quantum

Information: Measuring the Average Fidelity Decay". Physical Review Letters 92 (17): 177906. Bibcode 2004PhRvL..92q7906P.

doi: 10.1 103/PhysRevLett.92.177906.
[14] Poulin, David. "A Rough Guide to Quantum Chaos" (http://www.iqc.ca/publications/tutorials/chaos.pdf) (PDF). .
[15] Peres, A. (1995). Quantum Theory: Concepts and Methods. Dordrecht: Kluwer Academic.
[16] Lee, Jae-Seung; Khitrin, A. K. (2004). "Quantum amplifier: Measurement with entangled spins". Journal of Chemical Physics 121 (9):

3949. doi: 10. 1063/1. 1788661.

Butterfly effect

315

Further reading

• Devaney, Robert L. (2003). Introduction to Chaotic Dynamical Systems. Westview Press. ISBN 0813340853.

• Hilborn, Robert C. (2004). "Sea gulls, butterflies, and grasshoppers: A brief history of the butterfly effect in
nonlinear dynamics". American Journal of Physics 72 (4): 425—427. doi:10.1 119/1.1636492.

External links

• The meaning of the butterfly: Why pop culture loves the 'butterfly effect,' and gets it totally wrong (http://www.
boston. com/bostonglobe/ideas/articles/2008/06/08/the_meaning_of_the_butterfly/?page=full), Peter
Dizikes, Boston Globe, June 8, 2008

• From butterfly wings to single e-mail (http://www.news.cornell.edu/releases/Feb04/AAAS.Kleinberg.ws.
html) (Cornell University)

• New England Complex Systems Institute - Concepts: Butterfly Effect (http://necsi.edu/guide/concepts/
butterflyeffect. html)

• The Chaos Hypertextbook (http://hypertextbook.com/chaos/). An introductory primer on chaos and fractals.

• Weisstein, Eric W., " Butterfly Effect (http://mathworld.wolfram.com/ButterflyEffect.html)" from
MathWorld.

Standard map

The standard map is an area-preserving chaotic
map from a square with side 27r on t° itself. It is
defined by:

Standard map

316

Standard map

317

Orbits of the standard map for K = 2.0. The large green region is the main
chaotic region of the map.

A single orbit of the standard map for K=2.0. Magnified close-up centered at

6 = 0.282- P = 0o6 °. of total width/height 0.02. Note the extremely

uniform distribution of the orbit.

Pn+i =p n + K sin(0„)

0n-|-l = 8n + Pn+1

where p n and $ n are taken modulo 27r ■

This map is also known as the Chirikov— Taylor map or as the Chirikov standard map, The properties of chaos of

the standard map were established by Boris Chirikov in 1969. See more details at Scholarpedia entry . The

Standard map 318

quantized map is also known as the kicked rotator in the quantum chaos community.

Physical model

This map describes the motion of a simple mechanical system known as the kicked rotator. It consists of a stick that
is free of the gravitational force, which can rotate frictionlessly in a plane around an axis located in one of its tips,
and which is periodically kicked on the other tip. The variables # n and p n respectively determine the angular
position of the stick and its angular momentum after the n-th kick. The constant K measures the intensity of the
kicks.

The kicked rotator approximates systems studied in the fields of mechanics of particles, accelerator physics, plasma
physics, and solid state physics. For example, circular particle accelerators accelerate particles by applying periodic
kicks, as they circulate in the beam tube. Thus, the structure of the beam can be approximated by the kicked rotor.
However, this map is interesting from a fundamental point of view in physics and mathematics because it is a very
simple model of a conservative system that displays Hamiltonian chaos. It is therefore useful to study the
development of chaos in this kind of system.

Main properties

For K = Qthe map is linear and only periodic and quasiperiodic orbits are possible. When plotted in phase space
(the 6- p plane), periodic orbits appear as closed curves, and quasiperiodic orbits as necklaces of closed curves
whose centers lie in another larger closed curve. Which type of orbit is observed depends on the map's initial
conditions.

Nonlinearity of the map increases with K, and with it the possibility to observe chaotic dynamics for appropriate
initial conditions. This is illustrated in the figure, which displays a collection of different orbits allowed to the
standard map for various values of K > 0. All the orbits shown are periodic or quasiperiodic, with the exception
of the green one that is chaotic and develops in a large region of phase space as an apparently random set of points.
Particularly remarkable is the extreme uniformity of the distribution in the chaotic region, although this can be
deceptive: even within the chaotic regions, there are an infinite number of diminishingly small islands that are never
visited during iteration, as shown in the close-up.

Circle map

The standard map is related to the circle map, which has a single, similar iterated equation:

B n+1 =e n + n-Kan(9 n )

as compared to

0n+i = n +P n + Kam(0 n )

Pn+1 = B n+ i — 8 n

for the standard map, the equations reordered to emphasize similarity. In essence, the circle map forces the
momentum to a constant.

Standard map

319

References

[1] http://www.scholarpedia.org/article/Chirikov_standard_map

• Chirikov, B.V.. Research concerning the theory of nonlinear resonance and stochasticity. Preprint N 267,
Institute of Nuclear Physics, Novosibirsk (1969) (in Russian) [Engl. Transl., CERN Trans. 71 - 40, Geneva,
October (1971), Translated by A.T.Sanders], link (http://www.quantware.ups-tlse.fr/chirikov/refs/chil969e.
pdf)

• Chirikov, B.V.. A universal instability of many -dimensional oscillator systems. Phys. Rep. v. 52. p. 263 (1979)
Elsvier, Amsterdam.

• Lichtenberg, A.J. and Lieberman, M.A. (1992). Regular and Chaotic Dynamics. Springer, Berlin. ISBN
978-0-387-97745-4. Springer link (http://www.springer.com/math/analysis/book/978-0-387-97745-4)

• Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press New, York. ISBN 0-521-01084-5.

• Sprott, Mien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9.

External links

• Standard map (http://mathworld.wolfram.com/StandardMap.html) at Math World

• Chirikov standard map (http://www.scholarpedia.org/article/Chirikov_standard_map) at Scholarpedia (http://
www.scholarpedia.org)

• Website dedicated to Boris Chirikov (http://www.quantware.ups-tlse.fr/chirikov/)

• Interactive Java Applet visualizing orbits of the Standard Map (http://complexity.xozzox.de/nonlinmappings.
html), by Achim Luhn

Henon map

The Henon map is a discrete-time dynamical system. It
is one of the most studied examples of dynamical
systems that exhibit chaotic behavior. The Henon map
takes a point (x , y ) in the plane and maps it to a new
point

^7i+l — Vn + 1
Vtl+1 = bx n .

ax.

Henon map

320

The map depends on two parameters, a and b, which for the canonical Henon map have values of a = 1.4 and
b = 0.3. For the canonical values the Henon map is chaotic. For other values of a and b the map may be chaotic,
intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter
values may be obtained from its orbit diagram.

The map was introduced by Michel Henon as a simplified model of the Poincare section of the Lorenz model. For
the canonical map, an initial point of the plane will either approach a set of points known as the Henon strange
attractor, or diverge to infinity. The Henon attractor is a fractal, smooth in one direction and a Cantor set in another.
Numerical estimates yield a correlation dimension of 1.25 ± 0.02 and a Hausdorff dimension of 1.261 ± 0.003
for the attractor of the canonical map.

As a dynamical system, the canonical Henon map is interesting because, unlike the logistic map, its orbits defy a
simple description.

Attractor

The Henon map maps two points into themselves: these
are the invariant points. For the canonical values of a
and b of the Henon map, one of these points is on the
attractor:

x = 0.631354477... and y = 0.189406343...

This point is unstable. Points close to this fixed point
and along the slope 1.924 will approach the fixed point
and points along the slope -0.156 will move away from
the fixed point. These slopes arise from the
linearizations of the stable manifold and unstable
manifold of the fixed point. The unstable manifold of
the fixed point in the attractor is contained in the
strange attractor of the Henon map.

The Henon map does not have a strange attractor for all
values of the parameters a and b. For example, by
keeping b fixed at 0.3 the bifurcation diagram shows
that for a = 1.25 the Henon map has a stable periodic
orbit as an attractor.

Cvitanovic et al. have shown how the structure of the Henon strange attractor can be understood in terms of unstable
periodic orbits within the attractor.

0.5

~^?^

r»

-- ,___■

c|flH

^*V^ "

ra ^j^a-«y"

x 0.0 _

y ^k

*~ — -^

^~^~~~~~~

-0.5

1 1 1 1 1 1 1 1 1

1.0 1.1 1.2 1.3 1.4 1.5
a

Orbit diagram for the Henon map with b=0.3. Higher density

indicates increased probability of the y variable acquiring that value

for the given value of the a parameter.

Decomposition

The Henon map may be decomposed into an area-preserving bend:

{xi,yi) = {x,l -ax 2 +y),

a contraction in the x direction:

(^2,2/2) = (bx 1 ,y 1 ),

and a reflection in the line y = x:

(z*,3fe) = (s/2,z 2 )-

Henon map 321

References

[1] P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica 9D (1-2): 189—208.

Bibcode 1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-l.
[2] D.A. Russel, J.D. Hanson, and E. Ott (1980). "Dimension of strange attractors". Physical Review Letters 45 (14): 1 175.

Bibcode 1980PhRvL..45.1175R. doi:10.1103/PhysRevLett.45.1175.

• M. Henon (1976). "A two-dimensional mapping with a strange attractor". Communications in Mathematical
Physics 50 (1): 69-77. doi:10.1007/BF01608556.

• Predrag Cvitanovic, Gemunu Gunaratne, and Itamar Procaccia (1988). "Topological and metric properties of
Henon-type strange attractors". Physical Review A 38 (3): 1503-1520. doi:10.1103/PhysRevA.38.1503.
PMID 9900529.

• M. Michelitsch and O. E. Rossler (1989). "A New Feature in Henon's Map" (http://mathworld.wolfram.com/
HenonMap.html). Computers and Graphics 13 (2): 263-265. doi:10.1016/0097-8493(89)90070-8.. Reprinted in:
Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A.
Pickover). Amsterdam, Netherlands: Elsevier, pp. 69—71, 1998

External links

• Interactive Henon map (http://ibiblio.org/e-notes/Chaos/henon.htm) and Henon attractor (http://ibiblio.org/
e-notes/Chaos/strange.htm) in Chaotic Maps (http://ibiblio.org/e-notes/Chaos/contents.htm)

• Another interactive iteration of the Henon Map (http://complexity.xozzox.de/nonlinmappings.html) by A.
Luhn

• Orbit Diagram of the Henon Map (http://demonstrations.wolfram.eom/OrbitDiagramOfTheHenonMap//) by
C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr, The Wolfram Demonstrations Project.

Horseshoe map

In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into
itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while
studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by
squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a
horseshoe.

Most points eventually leave the square under the action of the map. They go to the side caps where they will, under
iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration
form a fractal set and are part of the invariant set of the map.

Horseshoe map

322

The squishing, stretching and folding of the horseshoe map are the essential elements that must be present in a
chaotic system. In the horseshoe map the squeezing and stretching are uniform. They compensate each other so that
the area of the square does not change. The folding is done neatly, so that the orbits that remain forever in the square
can be simply described.

For a horseshoe map:

• there are an infinite number of periodic orbits;

• periodic orbits of arbitrarily long period exist;

• the number of periodic orbits grows exponentially with the period; and

• close to any point of the fractal invariant set there is a point of a periodic orbit.

The horseshoe map

The horseshoe map /is a diffeomorphism defined from a region gof the plane into itself. The region gis a

square capped by two semi-disks. The action of /is defined through the composition of three geometrically defined

transformations. First the square is contracted along the vertical direction by a factor a < 1/2- The caps are

contracted so as to remain semi-disks attached to the resulting rectangle. Contracting by a factor smaller than one

half assures that there will be a gap between the branches of the horseshoe. Next the rectangle is stretched

horizontally by a factor of \ja\ the caps remain unchanged. Finally the resulting strip is folded into a

horseshoe-shape and placed back into Q .

The interesting part of the dynamics is the image of the square into itself. Once that part is defined, the map can be

extended to a diffeomorphism by defining its action on the caps. The caps are made to contract and eventually map

inside one of the caps (the left one in the figure). The extension of / to the caps adds a fixed point to the

non-wandering set of the map. To keep the class of horseshoe maps simple, the curved region of the horseshoe

should not map back into the square.

The horseshoe map is one-to-one: any point in the domain has a unique image, even though not all points of the
domain are the image of a point. The inverse of the horseshoe map, denoted by f cannot have as its domain the
entire region S, instead it must be restricted to the image of S under/, that is, the domain of/ isf(S).

By folding the contracted and stretched square in different ways, other types of horseshoe maps are possible.

(

The contracted square cannot overlap itself to assure that it remains one-to-one. When the action on the square is
extended to a diffeomorphism, the extension cannot always be done on the plane. For example, the map on the right
needs to be extended to a diffeomorphism of the sphere by using a "cap" that wraps around the equator.

The horseshoe map is an Axiom A diffeomorphism that serves as a model for the general behavior at a transverse
homoclinic point, where the stable and unstable manifolds of a periodic point intersect.

Horseshoe map

323

Dynamics of the map

The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given
periodic orbit. The neighborhood is chosen to be a small disk perpendicular to the orbit. As the system evolves,
points in this disk remain close to the given periodic orbit, tracing out orbits that eventually intersect the disk once
again. Other orbits diverge.

The behavior of all the orbits in the disk can be determined by considering what happens to the disk. The intersection
of the disk with the given periodic orbit comes back to itself every period of the orbit and so do points in its
neighborhood. When this neighborhood returns, its shape is transformed. Among the points back inside the disk are
some points that will leave the disk neighborhood and others that will continue to return. The set of points that never
leaves the neighborhood of the given periodic orbit form a fractal.

A symbolic name can be given to all the orbits that remain in the neighborhood. The initial neighborhood disk can be
divided into a small number of regions. Knowing the sequence in which the orbit visits these regions allows the orbit
to be pinpointed exactly. The visitation sequence of the orbits provide a symbolic representation of the dynamics,
known as symbolic dynamics.

Orbits

It is possible to describe the behavior of all initial conditions of the horseshoe map. An initial point u

[x, y } gets

mapped into the point u = /(m ). Its iterate is the point u =f(u,) =f (w ), and repeated iteration generates the orbit

M 0' M 1' M 2'-

Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in the left cap. This is because
the horseshoe maps the left cap into itself by an affine transformation, which has exactly one fixed point. Any orbit
that lands on the left cap never leaves it and converges to the fixed point in the left cap under iteration. Points in the
right cap get mapped into the left cap on the next iteration, and most points in the square get mapped into the caps.
Under iteration, most points will be part of orbits that converge to the fixed point in the left cap, but some points of
the square never leave.

Iterating the square

Under forward iterations of the horseshoe map, the
original square gets mapped into a series of horizontal
strips. The points in these horizontal strips come from
vertical strips in the original square. Let jSgbe the
original square, map it forward n times, and consider only
the points that fall back into the square S , which is a set
of horizontal stripes

H n = f n (S ) n So-
The points in the horizontal stripes came from the vertical
stripes

Pre-images of the square region

which are the horizontal strips ff mapped backwards n times. That is, a point in V will, under n iterations of the
horseshoe, end up in the set H n of vertical strips.

Horseshoe map

324

Invariant set

If a point is to remain indefinitely in the square, then it must belong to a set y^that
maps to itself. Whether this set is empty or not has to be determined. The vertical
strips Vjmap into the horizontal strips Hi, but not all points of Vjmap back into
V\- Only the points in the intersection of V^and i^may belong to y^, as can be
checked by following points outside the intersection for one more iteration.
The intersection of the horizontal and vertical stripes, H n r\ V n , are squares that
converge in the limit n — > °° to the invariant set /^. The structure of this set can be
better understood by introducing a system of labels for all the intersections — a
symbolic dynamics.

INI II

^M^a^H^M^M^HM ^H

Intersections that converge to the
invariant set

Example of an invariant measure

Symbolic dynamics

The intersection H n n V^,is contained in Vi- So any point that is in y\under
iteration must land in the left vertical strip A of V±, or on the right vertical strip B.
The lower horizontal strip of iJ^is the image of A and the upper horizontal strip is
the image of B, so H =f(A) u f(B). The strips A and B can be used to label the four
squares in the intersection of V^and H\ '■

\. A =f(A)nA \. B =f(A)cB
\. A =f(B)nA \. B =f(B)cB

The basic domains of the
horseshoe map

The set A consist of points from strip A that were in strip B in the previous iteration. A dot is used to separate the
region the point of an orbit is in from the region the point came from.

Horseshoe map 325

The notation can be extended to higher iterates of the horseshoe map. The vertical strips can be named according to
the sequence of visits to strip A or strip B. For example, the set ABB C V consists of the points from A that will all
land in B in one iteration and remain in B in the iteration after that:

ABB = { x € A \f(x) € B andf 2 (x) € B }

Working backwards from that trajectory determines a small region, the set ABB, within V .

The horizontal strips are named from their vertical strip pre-images. In this notation, the intersection of V and H
consists of 16 squares, one of which is

A AB.BB=f 2 < AB ^ BB -

All the points in A are in B and will continue to be in B for at least one more iteration. Their previous

trajectory before landing in BB was A followed by B.

Periodic orbits

Any one of the intersections A of a horizontal strip with a vertical strip, where P and F are sequences of As and
Bs, is an affine transformation of a small region in V . If P has k symbols in it, and iff (A ) and A intersect,
the region A will have a fixed point. This happens when the sequence P is the same as F. For example,
A „,„CV,nff, has at least one fixed point. This point is also the same as the fixed point in A,_ ._. By

ABAB-ABAB 4 4 F F F AB-AB J

including more and more ABs in the P and F part of the label of intersection, the area of the intersection can be made
as small as needed. It converges to a point that is part of a periodic orbit of the horseshoe map. The periodic orbit can
be labeled by the simplest sequence of As and Bs that labels one of the regions the periodic orbit visits.

For every sequence of As and Bs there is a periodic orbit.

References

• Stephen Smale (1967). "Differentiable dynamical systems". Bulletin of the American Mathematical Society 73
(6): 747-817. doi:10.1090/S0002-9904-1967-l 1798-1.

• P. Cvitanovic, G. Gunaratne, and I. Procaccia (1988). "Topological and metric properties of Henon-type strange
attractors". Physical Review A 38 (3): 1503-1520. doi:10.1103/PhysRevA.38.1503. PMID 9900529.

• Andre de Carvalho (1999). "Pruning fronts and the formation of horseshoes". Ergodic theory and dynamical
systems 19 (4): 851-894. doi:10.1017/S0143385799133972.

• Andre de Carvalho and Toby Hall (2002). "How to prune a horseshoe". Nonlinearity 15 (3): R19— R68.
doi:10.1088/0951-7715/15/3/201.

External links

• Smale Horseshoe at Scholarpedia.

• Interactive Smale horseshoe with Java applet and comments

References

[1] http://www.scholarpedia.org/article/Smale_Horseshoe
[2] http://www.ibiblio.org/e-notes/Chaos/homoclinic.htm

Coupled map lattice 326

Coupled map lattice

A coupled map lattice (CML) is a dynamical system that models the behavior of non-linear systems (especially
partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially
extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of
freedom diverge as the size of the system increases . Features of the CML are discrete time dynamics, discrete
underlying spaces (lattices or networks), and real (number or vector), local, continuous state variables . Studied

systems include populations, chemical reactions, convection, fluid flow and biological networks. Even recently,

Mi
CMLs have been applied to computational networks identifying detrimental attack methods and cascading

failures.

CML's are comparable to cellular automata models in terms of their discrete features . However, the value of each
site in a cellular automata network is strictly dependent on its neighbor (s) from the previous time step. Each site of
the CML is only dependent upon its neighbors relative to the coupling term in the recurrence equation. However, the
similarities can be compounded when considering multicomponent dynamical systems.

Introduction

The modeling of a CML generally incorporates a system of equations (coupled or uncoupled), a finite number of
variables, a global or local coupling scheme and the corresponding coupling terms. The dimension of the underlying
lattice can exist in infinite dimensions, but for this observation we restrict the lattice to two. Mappings of interest in
CMLs generally demonstrate a chaotic behavior. Such maps can be found here: List of chaotic maps.

A logistic mapping demonstrates chaotic behavior, easily identifiable in one dimension for parameter r > 3.57 (see
Logistic map). It is graphed across a small lattice and decoupled with respect to neighboring sites. The recurrence
equation is homogeneous, albeit randomly seeded. The parameter r is updated every time step (see Figure 1, Enlarge,
Summary):

■En+1 — rX n [l X n J

The result is a raw form of chaotic behavior in a map lattice. The range of the function is bounded so similar
contours through the lattice is expected. However, there are no significant spatial correlations or pertinent fronts to
the chaotic behavior. No obvious order is apparent.

For a basic coupling, we consider a 'single neighbor' coupling where the value at any given site sis mapped
recursively with respect to itself and the neighboring site g — 1- The coupling parameter ^ = 0.5i s equally
weighted.

x n+1 = (e)[rz„(l - x n )] a + {1 - e)[rz„(l - z„)] B _i
Even though each native recursion is chaotic, a more solid form develops in the evolution. Elongated convective
spaces persist throughout the lattice (see Figure 2).

Coupled map lattice

327

•■^^v^ Itiv^

^^^

^^^a>jp

^feg^ '->— g- - ^P

_- — ^_. ^ ■ ^

J ~ "~"^g^<

^^^^^ ^e

0.8
0.7
0.6
0.5
0.4
- 0.3

0.2

Figure 1 : An uncoupled logistic map lattice
with random seeding over forty iterates.

Figure 2: A CML with a single-neighbor
coupling scheme taken over forty iterates.

History

CMLs were first introduced in the mid 1980's through a series of closely released publications . Kapral

used CMLs for modeling chemical spatial phenomena. Kuznetsov sought to apply CMLs to electrical circuitry by
developing a renormalization group approach (similar to Feigenbaum's universality to spatially extended systems).

Kaneko's focus was more broad and he is still known as the most active researcher in this area
examined CML model was introduced by Kaneko in 1983 where the recurrence equation is as follows:

[10]

.The most

t+1

(1 - E)f(ul) + - (f(ul +l ) + f(ul_ x )) t G N, e G [0, 1]

where u' f8 an d /is a real mapping.
The applied CML strategy was as follows:

• Choose a set of field variables on the lattice at a macroscopic level. The dimension (not limited by the CML
system) should be chosen to correspond to the physical space being researched.

• Decompose the process (underlying the phenomena)into independent components.

• Replace each component by a nonlinear transformation of field variables on each lattice point and the coupling
term on suitable, chosen neighbors.

• Carry out each unit dynamics ("procedure") successively.

Classification

The CML system evolves through discrete time by a mapping on vector sequences. These mappings are a recursive
function of two competing terms: an individual nonlinear reaction, and a spatial interaction (coupling) of variable
intensity. CMLs can be classified by the strength of this coupling parameter(s).

Much of the current published work in CMLs is based in weak coupled systems where diffeomorphism of the
state space close to identity are studied. Weak coupling with monotonic (bistable) dynamical regimes demonstrate
spatial chaos phenomena and are popular in neural models . Weak coupling unimodal maps are characterized by
their stable periodic points and are used by genetic regulatory network models. Space-time chaotic phenomena can
be demonstrated from chaotic mappings subject to weak coupling coefficients and are popular in phase transition
phenomena models.

Intermediate and strong coupling interactions are less prolific areas of study. Intermediate interactions are studied
with respect to fronts and traveling waves, riddled basins, riddled bifurcations, clusters and non-unique phases.

Coupled map lattice

328

Strong coupling interactions are most well known to model synchronization effects of dynamic spatial systems such
as the Kuramoto model.

ri3i

These classifications do not reflect the local or global (GMLs )coupling nature of the interaction. Nor do they

ri4i

consider the frequency of the coupling which can exist as a degree of freedom in the system . Finally, they do not
distinguish between sizes of the underlying space or boundary conditions.

Surprisingly the dynamics of CMLs have little to do with the local maps that constitute their elementary components.
With each model a rigorous mathematical investigation is needed to identify a chaotic state (beyond visual
interpretation). Rigorous proofs have been performed to this effect. By example: the existence of space-time chaos in
weak space interactions of one-dimensional maps with strong statistical properties was proven by Bunimovich and
Sinai in 1988 . Similar proofs exist for weakly hyperbolic maps under the same conditions.

Unique CML qualitative classes

CMLs have revealed novel qualitative universality classes in (CML) phenomenology. Such classes include:

Spatial bifurcation and frozen chaos

Pattern Selection

Selection of zig-zag patterns and chaotic diffusion of defects

Spatio-temporal intermittency

Soliton turbulence

Global traveling waves generated by local phase slips

Spatial bifurcation to down-flow in open flow systems.

Visual phenomena

The unique qualitative classes listed above can be visualized. By applying the Kaneko 1983 model to the logistic
f(x n ) = 1 — aa; 2ma P> several of the CML qualitative classes may be observed. These are demonstrated below,
note the unique parameters:

Pattern Selection

Chaotic Brownian Motion of Defect

Figure 1: Sites are divided into non-uniform clusters,
where the divided patterns are regarded as attractors.
Sensitivity to initial conditions exist relative to a
<1.5.

Defect Turbulence

Figure 2: Near uniform sized clusters (a - 1.71, £
= 0.4).

Spatiotemporal Intermittency I

Figure 3: Deflects exist in the system and fluctuate
chaotically akin to Brownian motion (a = 1.85, 8
= 0.1).

Spatiotemporal Intermittency II

Coupled map lattice

329

Figure 4: Many defects are generated and turbulently
collide (a =1.895, £ = 0.1).

Fully Developed Spatiotemporal Chaos

Figure 5 : Each site transits between a coherent state
and chaotic state intermittently (a = 1.75, e = 0.6),
Phase I.

Traveling Wave

Figure 6: The coherent state, Phase II.

Figure 7: Most sites independently oscillate
chaotically (a = 2.00, e = 0.3).

Figure 8: The wave of clusters travels at 'low' speeds
(a = 1.47, 8 = 0.5).

Quantitative analysis quantifiers

Coupled map lattices being a prototype of spatially extended systems easy to simulate have represented a benchmark
for the definition and introduction of many indicators of spatio-temporal chaos, the most relevant ones are

The power spectrum in space and time

Lyapunov spectra

Dimension density

Kolmogorov— Sinai entropy density

Distributions of patterns

Pattern entropy

Propagation speed of finite and infinitesimal disturbance

Mutual information and correlation in space-time

Lyapunov exponents, localization of Lyapunov vectors

Comoving and sub-space time Lyapunov exponents.

Spatial and temporal Lyapunov exponents

[17]

Coupled map lattice 330

References

[I] http://en.wiktionary.org/wiki/spatiotemporal

[2] Kaneko, Kunihiko. "Overview of Coupled Map Lattices." Chaos 2, Num3(1992): 279.

[3] Chazottes, Jean-Rene, and Bastien Fernandez. Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Springer,

2004. pgs 1-4
[4] Xu, Jian. Wang, Xioa Fan. " Cascading failures in scale-free coupled map lattices." IEEE International Symposium on Circuits and Systems "

ISCAS Volume 4, (2005): 3395-3398.
[5] R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press, Cambridge, England, 1997).
[6] K. Kaneko, Prog. Theor. Phys. 72, 480 (1984)
[7] I. waller and R. Kapral; Phys. Rev. A 30 2047 (1984)
[8] J. Crutchfield, Phyisca D 10, 229 (1984)

[9] S. P.Kuznetsov and A. S. Pikovsky, Izvestija VUS, Radiofizika 28, 308 (1985)
[10] http://chaos.cu-tokyo.ac.jp/

[II] Lectures from the school-forum (CML 2004) held in Paris, June 21{July 2, 2004. Edited by J.-R. Chazottes and B. Fernandez. Lecture Notes
in Physics, 671. Springer, Berlin (2005)

[12] Nozawa, Hiroshi. "A neural network model." Chaos 2, Num3(1992): 377.

[13] Ho, Ming-Ching. Hung, Yao-Chen. Jiang, I-Min. "Phase synchronization in inhomogenous globally coupled map lattices. Phyiscs Letter A.

324(2004)450-457. (http://www.phys.sinica.edu. tw/~statphys/publications/2004_full_text/M_C_Ho_PLA_324_450(2004).pdf)
[14] http://www.mat.uniroma2.it/~liverani/Lavori/live0803.pdf
[15] L.A. Bunimovich and Ya. G. Sinai. "Nonlinearity" Vol. 1 pg 491 (1988)
[16] Lyapunov Spectra of Coupled Map Lattices, S. Isola, A. Politi, S. Ruffo, and A. Torcini (http://www.fi.isc.cnr.it/users/antonio.politi/

Reprints/052. pdf)
[17] S. Lepri, A. Politi and A. Torcini Chronotopic Lyapunov Analysis: (I) a Detailed Characterization of ID Systems (http://xxx.lanl.gov/

abs/chao-dyn/9504005), J. Stat. Phys., 82 5/6 (1996) 1429.

Further reading

• Google Library (2005). Dynamics of Coupled Map Lattices (http://books.google.com/
books?id=a63Q8DhKA44C&dq=coupled+map+lattices&source=gbs_summary_s&hl=en). Springer.
ISBN 9783540242895. Archived from the original (http://books.google.com/?id=a63Q8DhKA44C&
dq=coupled+map+lattices) on 2008-03-29.

• Shawn D. Pethel, Ned J. Corron, and Erik Bollt. "Symbolic Dynamics of Coupled Map Lattices" (http://people.
clarkson.edu/~bolltem/Papers/PhysRevLett_96_034105PethelCorronBollt.pdf). Physical Review Letters.
Archived from the original (http://dx.dio.org/10.1103/PhysRevLett.96.034105) on 2008-03-29.

• E. Atlee Jackson, Perspectives of Nonlinear Dynamics: Volume 2 (http://books.google.com/
books?id=M2E0AAAAIAAJ&source=gbs_ViewAPI), Cambridge University Press, 1991, ISBN 0521426332

• H.G, Schuster and W. Just, Deterministic Chaos (http://www.whsmith.co.uk/CatalogAndSearch/
ProductDetails.aspx?productID=9783527404155), John Wiley and Sons Ltd, 2005, ISBN 3527404155

• Introduction to Chaos and Nonlinear Dynamics (http://brain.cc.kogakuin.ac.Jp/~kanamaru/Chaos/e/)

External links

• Kaneko Laboratory (http://chaos.cu-tokyo.ac.jp/)

• Institut Henri Poincare, Paris, June 21 — July 2, 2004 (http://www.cpht.polytechnique.fr/cpth/cml2004/)

• Istituto dei Sistemi Complessi (http://www.fi.isc.cnr.it/), Florence, Italy

Software

• Java CML/GML web-app (http://brain.cc.kogakuin.ac.Jp/~kanamaru/Chaos/e/CMLGCM/)

• AnT 4.669 — A simulation and Analysis Tool for Dynamical Systems (http://ant4669.de/)

List of chaotic maps

331

List of chaotic maps

In mathematics, a chaotic map is a map (= evolution function) that exhibits some sort of chaotic behavior. Maps may
be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated
functions. Chaotic maps often occur in the study of dynamical systems.

Chaotic maps often generate fractals. Although a fractal may be constructed by an iterative procedure, some fractals
are studied in and of themselves, as sets rather than in terms of the map that generates them. This is often because
there are several different iterative procedures to generate the same fractal.

List of chaotic maps

Map

Time
domain

Space
domain

Number of space
dimensions

Also known as

Arnold's cat map

discrete

real

Baker's map

discrete

real

Bogdanov map

Chossat-Goluhitsky symmetry map

Circle map

discrete

real

Cobweb map

Complex quadratic map

discrete

complex

Complex squaring map

discrete

complex

Complex Cubic map

Degenerate Double Rotor map

Double Rotor map

Duffing map

discrete

real

Duffing equation

continuous

real

Dyadic transformation

discrete

real

2x mod 1 map, Bernoulli map, doubling map,
sawtooth map

Exponential map

discrete

complex

Gauss map

discrete

real

mouse map, Gaussian map

Generalized Baker map

Gingerbreadman map

discrete

real

Gumowski/Mira map

Henon map

discrete

real

Henon with 5th order polynomial

Hitzl-Zele map

Horseshoe map

discrete

real

Ikeda map

discrete

real

Interval exchange map

discrete

real

Kaplan- Yorke map

discrete

real

Linear map on unit square

List of chaotic maps

332

Logistic map

discrete

real

Lorenz attractor

continuous

real

Lorenz system's Poincare Return map

Lozi map

Nordmark truncated map

Pomeau-Manneville maps for
intermittent chaos

discrete

real

land 2

Normal-form maps for intermittency (Types I,
and III)

Pulsed rotor

Quasiperiodicity map

Rabinovich-Fabrikant equations

continuous

real

Random Rotate map

Rossler map

continuous

real

Shobu-Ose-Mori piecewise-linear
map

discrete

real

piecewise-linear approximation for
Pomeau-Manneville Type I map

Sinai map - See [1]

Symplectic map

Standard map, Kicked rotor

discrete

real

Chirikov standard map, Chirikov-Taylor map

Tangent map

Tent map

discrete

real

Tinkerbell map

discrete

real

Triangle map

Van der Pol oscillator

continuous

real

Zaslavskii map

discrete

real

Zaslavskii rotation map

List of fractals

Cantor set

de Rham curve

Gravity set, or Mitchell-Green gravity set

Julia set - derived from complex quadratic map

Newton fractal

Nova fractal - derived from Newton fractal

Koch snowflake - special case of de Rham curve

Lyapunov fractal

Mandelbrot set - derived from complex quadratic map

Menger sponge

Sierpinski carpet

Sierpinski triangle

List of chaotic maps

333

References

[1] http://www.maths.ox.ac.uk/~mcsharry/papers/dynsysl8n3pl91y2003mcsharry.pdf

Chua's circuit

Chua's circuit is a simple electronic
circuit that exhibits classic chaos
theory behavior. It was introduced in
1983 by Leon O. Chua, who was a
visitor at Waseda University in Japan
at that time. The ease of construction
of the circuit has made it a ubiquitous
real-world example of a chaotic
system, leading some to declare it "a

paradigm for chaos

.,[2]

GND

A version of Chua's circuit without Chua's Diode

Chaotic criteria

An autonomous circuit made from standard components (resistors, capacitors, inductors) must satisfy three criteria
before it can display chaotic behaviour. It must contain:

1 . one or more nonlinear elements

2. one or more locally active resistors

3. three or more energy-storage elements.

Chua's circuit is the simplest electronic circuit meeting these criteria. As shown in the figure, the energy storage
elements are two capacitors (labeled CI and C2) and an inductor (labeled LI). There is an active resistor (labeled R).
There is a nonlinear resistor made of two linear resistors and two diodes. At the far right is a negative impedance
converter made from three linear resistors and an operational amplifier.

Model

By means of the application of the laws of electromagnetism, the dynamics of Chua's circuit can be accurately
modeled by means of a system of three nonlinear ordinary differential equations in the variables x(t), y(t) and z(t),
which give the voltages across the capacitors CI and C2, and the intensity of the electrical current in the inductor LI,
respectively. These equations read:

~dt
dy

dt
dz

~dl

The function f(x) describes the electrical response of the nonlinear resistor, and its shape depends on the particular

configuration of its components. The parameters a and |3 are determined by the particular values of the circuit
components.

A chaotic attractor, known as "The Double Scroll" because of its shape in the (x,y,z) space, was first observed in a
circuit containing a nonlinear element such that f(x) was a 3-segment piecewise-linear function

a[y-x- f(x)]
= x —y + z

-Py

[3]

Chua's circuit

334

The easy experimental implementation of the circuit, combined with the existence of a simple and accurate
theoretical model, makes Chua's circuit a useful system to study many fundamental and applied issues of chaos
theory. Because of this, it has been object of much study, and appears widely referenced in the literature.

References

[1] Matsumoto, Takashi (December 1984). "A Chaotic Attractor from Chua's Circuit" (http://www.eecs.berkeley.edu/~chua/papers/
Matsumoto84.pdf). IEEE Transactions on Circuits and Systems (IEEE) CAS-31 (12): 1055-1058. . Retrieved 2008-05-01.

[2] Madan, Rabinder N. (1993). Chua's circuit: a paradigm for chaos. River Edge, N.J.: World Scientific Publishing Company.
ISBN 9810213662.

[3] Chua, Leon O.; Matsumoto, T., and Komuro, M. (August 1985). "The Double Scroll" (http://ieeexplore.ieee.org/iel5/31/23571/
01085791.pdf). IEEE Transactions on Circuits and Systems (IEEE) CAS-32 (8): 798-818. . Retrieved 2008-05-01.

Books

• Chaos synchronization in Chua's circuit, Leon O Chua, Berkeley : Electronics Research Laboratory, College of
Engineering, University of California, [1992], OCLC: 44107698

External links

• Chua's Circuit: Diagram and discussion (http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html)

• NOEL laboratory. Leon O. Chua's laboratory at the University of California, Berkeley (http://nonlinear.eecs.
berkeley.edu)

• References (http://www.eecs.berkeley.edu/~chua/circuitrefs.html)

• Chua and Memristors (http://blog.wired.com/gadgets/2008/04/scientists-prov.html)

Double pendulum

In mathematics, in the area of dynamical systems, a double pendulum
is a pendulum with another pendulum attached to its end, and is a
simple physical system that exhibits rich dynamic behavior with a
strong sensitivity to initial conditions. The motion of a double
pendulum is governed by a set of coupled ordinary differential
equations. For certain energies its motion is chaotic.

Analysis

Several variants of the double pendulum may be considered; the two
limbs may be of equal or unequal lengths and masses, they may be
simple pendulums or compound pendulums (also called complex
pendulums) and the motion may be in three dimensions or restricted to
the vertical plane. In the following analysis, the limbs are taken to be
identical compound pendulums of length £ and mass m , and the
motion is restricted to two dimensions.

///////////

A double pendulum consists of two pendulums
attached end to end.

Double pendulum

335

.(0, Q)

Double compound pendulum.

In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the centre of
mass of each limb is at its midpoint, and the limb has a moment of inertia of J = -^mf 2 about that point.

It is convenient to use the angle between each limb and the vertical as the generalized coordinates defining the
configuration of the system. These angles are denoted 6 and 6 . The position of the centre of mass of each rod may
be written in terms of these two coordinates. If the origin of the Cartesian coordinate system is taken to be at the
point of suspension of the first pendulum, then the centre of mass of this pendulum is at:

x 1

2/1

sin 0i,

COS 01

and the centre of mass of the second pendulum is at
x 2 = £ ( sin &i + - sin 2 J ,

-£ [ cos &i + - COS 8 2

This is enough information to write out the Lagrangian.

Lagrangian

The Lagrangian is

L = Kinetic Energy — Potential Energy

= -m {vl + vl) + -I (e\ + 0(j - mg { yi + y 2 )

= -m (±1 + y\ + ± 2 2 + yl) + -I (df + 2 2 ) - mg ( Vl + y 2 )

The first term is the linear kinetic energy of the center of mass of the bodies and the second term is the rotational
kinetic energy around the center of mass of each rod. The last term is the potential energy of the bodies in a uniform
gravitational field. The dot-notation indicates the time derivative of the variable in question.

Substituting the coordinates above and rearranging the equation gives

L = -mt
6

0\ + 40* + 30\0 2 cos(0i - 2 ) + -mg£ (3 cos 0i + cos 2 ) .

Double pendulum

336

There is only one conserved quantity (the energy), and no
conserved momenta. The two momenta may be written as

Motion of the double compound pendulum

(from numerical integration of the equations of

motion)

Long exposure of double pendulum exhibiting chaotic
motion (tracked with an LED)

POi.

ae 1

\me

80i + 30 2 cos(0i - 2 )

and

Pe 2

-mt

20 2 + 3fJx cos(0i - 2 )

dL _

d6 2 ~ 6'
These expressions may be inverted to get

6 2p 9l - 3 cos (0i - 2 )£>0 2

and

02 =

mi 2 16-9cos 2 (0i-0 2 )
6 8p 92 - 3 cos(0i - 02)^8!

mi 2 16 -9 cos 2 (0i -0 2 )
The remaining equations of motion are written as

and

PBj.

P9 2

dL
df 2

--m£ 2
2

-m£ 2
2

0i02sin(0i-0 2 ) + 3|sin0i

-0i0 2 sm(0i-0 2 ) + |sin0 2

These last four equations are explicit formulae for the time evolution of the system given its current state. It is not
possible to go further and integrate these equations analytically, to get formulae for 6 and 6 as functions of time. It
is however possible to perform this integration numerically using the Runge Kutta method or similar techniques.

Double pendulum 337

Chaotic motion

Graph of the time for the pendulum to flip over as
a function of initial conditions

The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions. The image
to the right shows the amount of elapsed time before the pendulum "flips over," as a function of initial conditions.
Here, the initial value of 6 ranges along the ^-direction, from -3 to 3. The initial value 6 ranges along the
^'-direction, from -3 to 3. The colour of each pixel indicates whether either pendulum flips within lOi/f la (g reen )>

within 100i fifg (red), lOOOi/fTff (P ur P le ) or 10000i/£/g(blue). Initial conditions that don't lead to a flip

within lOOOOi/f/g ^ e plotted white.

The boundary of the central white region is defined in part by energy conservation with the following curve:

3 cos 0i + cos 62 = 2.
Within the region defined by this curve, that is if

3 cos #i + cos 62 > 2,

then it is energetically impossible for either pendulum to flip. Outside this region, the pendulum can flip, but it is a
complex question to determine when it will flip.

The lack of a natural excitation frequency has led to the use of double pendulum systems in seismic resistance
designs in buildings, where the building itself is the primary inverted pendulum, and a secondary mass is connected
to complete the double pendulum.

References

• Meirovitch, Leonard (1986). Elements of Vibration Analysis (2nd edition ed.). McGraw-Hill
Science/Engineering/Math. ISBN 0-07-041342-8.

• Eric W. Weisstein, Double pendulum (2005), ScienceWorld (contains details of the complicated equations
involved) and "Double Pendulum by Rob Morris, Wolfram Demonstrations Project, 2007 (animations of those
equations).

• Peter Lynch, Double Pendulum , (2001). (Java applet simulation.)

• Northwestern University, Double Pendulum , (Java applet simulation.)

• Theoretical High-Energy Astrophysics Group at UBC, Double pendulum , (2005).

Double pendulum

338

External links

• Animations and explanations of a double pendulum and a physical double pendulum (two square plates) by
Mike Wheatland (Univ. Sydney)

ro]

• Video of a double square pendulum with three (almost) identical starting conditions.

Double pendulum physics simulation from www.myphysicslab.com

[10]

[9]

Simulation, equations and explanation of Rott's pendulum

Comparison videos of a double pendulum with the same initial starting conditions on YouTube

ri2i

Double Pendulum Simulator - An open source simulator written in C++ using the Qt tookit.
Online Java simulator of the Imaginary exhbibition.

References

[I] http://scienceworld.wolfram.com/physics/DoublePendulum.html
[2] http://demonstrations.wolfram.com/DoublePendulum/

[3] http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html

[4] http://www.physics.northwestern.edu/vpl/mechanics/pendulum.html

[5] http://tabitha.phas.ubc.ca/wiki/index.php/Double_pendulum

[6] http://www.physics.usyd.edu.au/~wheat/dpend_html/

[7] http://www.physics.usyd.edu.au/~wheat/sdpend/

[8] http://www.youtube.com/watch?v=Uzlccwt5SKc&NR=l

[9] http://www.myphysicslab.com/dbl_pendulum.html

[10] http://www.chris-j.co.uk/rott.php

[II] http://www.youtube.com/watch?v=02ySvbL3-yA
[12] http://freddie.witherden.org/tools/doublependulum/

[13] http://www.imaginary2008.de/cinderella/english/G2.html

Dynamical billiards

The Bunimovich stadium is a chaotic dynamical
billiard

A billiard is a dynamical system in which a particle alternates between
motion in a straight line and specular reflections from a boundary.
When the particle hits the boundary it reflects from it without loss of
speed. Billiard dynamical systems are Hamiltonian idealizations of the
game of billiards, but where the region contained by the boundary can
have shapes other than rectangular and even be multidimensional.
Dynamical billiards may also be studied on non-Euclidean geometries;
indeed, the very first studies of billiards established their ergodic
motion on surfaces of constant negative curvature. The study of
billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory.

The motion of the particle in the billiard is a straight line, with constant energy, between reflections with the
boundary (a geodesic if the Riemannian metric of the billiard table is not flat). All reflections are specular: the angle
of incidence just before the collision is equal to the angle of reflection just after the collision. The sequence of
reflections is described by the billiard map that completely characterizes the motion of the particle.

Billiards capture all the complexity of Hamiltonian systems, from integrability to chaotic motion, without the
difficulties of integrating the equations of motion to determine its Poincare map. Birkhoff showed that a billiard
system with an elliptic table is integrable.

Dynamical billiards 339

Equations of motion

The Hamiltonian for a particle of mass m moving freely without friction on a surface is:

S(R«)=|Uv(,),

where VY(jf)is a potential designed to be zero inside the region Qin which the particle can move, and infinity
otherwise:

™ = (° 9 M

I 00 q (fz \l.

This form of the potential guarantees a specular reflection on the boundary. The kinetic term guarantees that the
particle moves in a straight line, without any change in energy. If the particle is to move on a non-Euclidean
manifold, then the Hamiltonian is replaced by:

Am
where gij(q)is the metric tensor at point q £j O. Because of the very simple structure of this Hamiltonian, the

equations of motion for the particle, the Hamilton— J acobi equations, are nothing other than the geodesic equations on
the manifold: the particle moves along geodesies.

Notable billiard tables
Hadamard's billiards

Hadamard's billiards concern the motion of a free point particle on a surface of constant negative curvature, in
particular, the simplest compact Riemann surface with negative curvature, a surface of genus 2 (a two-holed donut).
The model is exactly solvable, and is given by the geodesic flow on the surface. It is the earliest example of
deterministic chaos ever studied, having been introduced by Jacques Hadamard in 1898.

Artin's billiard

Artin's billiard considers the free motion of a point particle on a surface of constant negative curvature, in particular,
the simplest non-compact Riemann surface, a surface with one cusp. It is notable for being exactly solvable, and yet
not only ergodic but also strongly mixing. It is an example of an Anosov system. This system was first studied by
Emil Artin in 1924.

Dynamical billiards

340

Sinai billiard

The table of the Sinai billiard is a square with a disk removed from its
center; the table is flat, having no curvature. The billiard arises from
studying the behavior of two interacting disks bouncing inside a
square, reflecting off the boundaries of the square and off each other.
By eliminating the center of mass as a configuration variable, the
dynamics of two interacting disks reduces to the dynamics in the Sinai
billiard.

The billiard was introduced by Yakov G. Sinai as an example of an
interacting Hamiltonian system that displays physical thermodynamic
properties: all of its possible trajectories are ergodic and it has a
positive Lyapunov exponent. As a model of a classical gas, the Sinai
billiard is sometimes called the Lorentz gas.

A trajectory in the Sinai billiard

Sinai's great achievement with this model was to show that the

classical Boltzmann— Gibbs ensemble for an ideal gas is essentially the maximally chaotic Hadamard billiards.

Bunimovich stadium

The table called the Bunimovich stadium is a rectangle capped by semicircles. Until it was introduced by Leonid
Bunimovich, billiards with positive Lyapunov exponents were thought to need convex scatters, such as the disk in
the Sinai billiard, to produce the exponential divergence of orbits. Bunimovich showed that by considering the orbits
beyond the focusing point of a concave region it was possible to obtain exponential divergence.

Generalized billiards

Generalized billiards (GB) describe a motion of a mass point (a particle) inside a closed domain TJ G ffi^with the
piece-wise smooth boundary p. On the boundary pthe velocity of point is transformed as the particle underwent
the action of generalized billiard law. GB were introduced by Lev D. Pustyl'nikov in the general case in (in
Notes), and, in the case when TT is a parallelepiped in (in Notes) in connection with the justification of the second
law of thermodynamics (the law of entropy increase). From the physical point of view, GB describe a gas consisting
of finitely many particles moving in a vessel, while the walls of the vessel heat up or cool down. The essence of the
generalization is the following. As the particle hits the boundary T\ its velocity transforms with the help of a given
function /('y, t), defined on the direct product p x ^(where Jjlis the real line, <y £j Tis a point of the
boundary and £ g: fl^is time), according to the following law. Suppose that the trajectory of the particle, which
moves with the velocity v, intersects pat the point -y £j Tat time £*, Then at time £*the particle acquires the
velocity y* , as if it underwent an elastic push from the infinitely-heavy plane p* , which is tangent to pat the
point "f , and at time f* moves along the normal to pat 7 with the velocity — — f'y t*\. We emphasize that the

dt y '' !

position of the boundary itself is fixed, while its action upon the particle is defined through the function f .

We take the positive direction of motion of the plane p* to be towards the interior of TJ . Thus if the derivative

—— (y t) > > m en the particle accelerates after the impact.
dt Kh '

If the velocity y* , acquired by the particle as the result of the above reflection law, is directed to the interior of the

domain TJ , then the particle will leave the boundary and continue moving in TJ until the next collision with p . If

the velocity .y*is directed towards the outside of TT> then the particle remains on pat the point "f until at some

time £ > £* the interaction with the boundary will force the particle to leave it.

Dynamical billiards 341

If the function /('y, i)does not depend on time £ , i.e., df/dt = 0, the generalized billiard coincides with the

classical one.

This generalized reflection law is very natural. First, it reflects an obvious fact that the walls of the vessel with gas
are motionless. Second the action of the wall on the particle is still the classical elastic push. In the essence, we
consider infinitesimally moving boundaries with given velocities.

It is considered the reflection from the boundary pboth in the framework of classical mechanics (Newtonian case)
and the theory of relativity (relativistic case).

Main results: in the Newtonian case the energy of particle is bounded, the Gibbs entropy is a constant, (in

Notes) and in relativistic case the energy of particle, the Gibbs entropy, the entropy with respect to the phase volume
grow to infinity, (in Notes), references to generalized billiards.

Quantum chaos

The quantum version of the billiards is readily studied in several ways. The classical Hamiltonian for the billiards,
given above, is replaced by the stationary-state Schrodinger equation Htj) = Elp or, more precisely,

n 2

—Z—&ij> n (q) = E n j> n (q),
zm

where A is the Laplacian. The potential that is infinite outside the region Qbut zero inside it translates to the
Dirichlet boundary conditions:

ip n (q) = for q <£ Q.
As usual, the wavefunctions are taken to be orthonormal:

/ ip m (q)ip n (q) dq = S mn .
Jn

Curiously, the free-field Schrodinger equation is the same as the Helmholtz equation,

(A + k 2 ) i) = 0,

with

. r. 2mE„

h 2 '

This implies that two and three-dimensional quantum billiards can be modelled by the classical resonance modes of a

radar cavity of a given shape, thus opening a door to experimental verification. (The study of radar cavity modes
must be limited to the transverse magnetic (TM) modes, as these are the ones obeying the Dirichlet boundary
conditions).

The semi -classical limit corresponds to fa — j. which can be seen to be equivalent to m — > OO , the mass
increasing so that it behaves classically.

As a general statement, one may say that whenever the classical equations of motion are integrable (e.g. rectangular
or circular billiard tables), then the quantum-mechanical version of the billiards is completely solvable. When the
classical system is chaotic, then the quantum system is generally not exactly solvable, and presents numerous
difficulties in its quantization and evaluation. The general study of chaotic quantum systems is known as quantum
chaos.

A particularly striking example of scarring on an elliptical table is given by the observation of the so-called quantum
mirage.

Dynamical billiards 342

Applications

The most practical application of theory of quantum billiards is related with double-clad fibers. In such a fiber laser,
the small core with low Numerical Aperture confines the signal, and the wide cladding confines the multi-mode
pump. In the paraxial approximation, the complex field of pump in the cladding behaves like a wave function in the
quantum billiard. The modes of the cladding with scarring may avoid the core, and symmetrical configurations
enhance this effect. The chaotic fibers provide good coupling; in the first approximation, such a fiber can be
described with the same equations as an idealized billiard. The coupling is especially poor in fibers with circular
symmetry while the spiral-shaped fiber — with the core close to the chunk of the spiral — shows good coupling
properties. The small spiral deformation forces all the scars to be coupled with the core.

Notes

[1] L. D. Pustyl'nikov, "The law of entropy increase and generalized billiards", Russ. Math. Surveys 54(3), pp. 650-651 (1999).

[2] L. D. Pustyl'nikov, "Poincare models, rogorous justification of the second law of thermodynamics from mechanics, and the Fermi

acceleration mechanism", Russ. Math. Surveys 50(1), pp. 145-189 (1995).
[3] L. D. Pustyl'nikov, "Generalized Newtonian periodic billiards in a ball", UMN, 60(2), pp. 171-172 (2005); English translation in Russian.

Math. Surveys, 60(2), pp. 365-366 (2005).
[4] Mikhail V. Deryabin and Lev D. Pustyl'nikov, "Nonequilibrium Gas and Generalized Billiards", Journal of statistical physics, 126(1), Januar,

pp. 117-132(2007).
[5] Leproux, P.; S. Fevrier, V. Doya, P. Roy, and D. Pagnoux (2003). "Modeling and optimization of double-clad fiber amplifiers using chaotic

propagation of pump" (http://www.ingentaconnect.com/content/ap/of/2001/00000007/00000004/art00361). Optical Fiber Technology 7

(4): 324-339. doi:10.1006/ofte.2001.0361. .
[6] Kouznetsov, D.; Moloney, J.V. (2004). "Boundary behavior of modes of Dirichlet Laplacian" (http://www.metapress.com/content/

be01ua88cwybywnl/?p=5464d03ba7e7440f9827207df673c804&pi=6). Journal of Modern Optics 51 (13): 1955-1962. .

References
Sinai's billiards

• Sinai, Ya. G. (1963). "[On the foundations of the ergodic hypothesis for a dynamical system of statistical
mechanics]" (in Russian). Doklady Akademii Nauk SSSR 153 (6): 1261—1264. (in English, Sov. Math Dokl. 4
(1963) pp. 1818-1822).

• Ya. G. Sinai, "Dynamical Systems with Elastic Reflections", Russian Math. Surveys, 25, (1970) pp. 137—191.

• V. I. Arnold and A. Avez, Theorie ergodique des systems dynamiques, (1967), Gauthier-Villars, Paris. (English
edition: Benjamin-Cummings, Reading, Mass. 1968). (Provides discussion and references for Sinai's billiards.)

• D. Heitmann, J. P. Kotthaus, "The Spectroscopy of Quantum Dot Arrays", Physics Today (1993) pp. 56—63.
(Provides a review of experimental tests of quantum versions of Sinai's billiards realized as nano-scale
(mesoscopic) structures on silicon wafers.)

• S. Sridhar and W. T. Lu, " Sinai Billiards, Ruelle Zeta-functions and Ruelle Resonances: Microwave Experiments
(http://sagar.physics.neu.edu/preprints/sinai-ruelle-jsp2002.pdf)", (2002) Journal of Statistical Physics, Vol.

108 Nos. 5/6, pp. 755-766.

• Linas Vepstas, Sinai's Billiards (http://www.linas.org/art-gallery/billiards/billiards.html), (2001). (Provides
ray-traced images of Sinai's billiards in three-dimensional space. These images provide a graphic, intuitive
demonstration of the strong ergodicity of the system.)

Dynamical billiards 343

Strange billiards

• T. Schiirmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A28, page 5033ff,
1995. PDF-Document (http://arxiv.org/abs/nlin/0208048)

Bunimovich stadium

• L. A. Bunimovich, "On the Ergodic Properties of Nowhere Dispersing Billiards", Commun Math Phys, 65 (1979)
pp. 295-312.

• L. A. Bunimovich and Ya. G. Sinai, "Markov Partitions for Dispersed Billiards", Commun Math Phys, 78 (1980)
pp. 247-280.

• Flash animation illustrating the chaotic Bunimovich Stadium (http://www.upscale.utoronto.ca/GeneralInterest/
Harrison/Flash/Chaos/Bunimovich/Bunimovich.html)

Generalized billiards

• M. V. Deryabin and L. D. Pustyl'nikov, "Generalized relativistic billiards", Reg. and Chaotic Dyn. 8(3),
pp. 283-296 (2003).

• M. V. Deryabin and L. D. Pustyl'nikov, "On Generalized Relativistic Billiards in External Force Fields", Letters
in Mathematical Physics, 63(3), pp. 195-207 (2003).

• M. V. Deryabin and L. D. Pustyl'nikov, "Exponential attractors in generalized relativistic billiards", Comm. Math.
Phys. 248(3), pp. 527-552 (2004).

External links

• Weisstein, Eric W., " Billiards (http://mathworld.wolfram.com/Billiards.html)" from MathWorld.

• Simulation of the Sinai Billiard (http://xweb.geos.ed.ac.uk/~stephan/mod_SinaiBilliard.en.html) (Stephan
Matthiesen)

Bifurcation theory

344

Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family,
such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most
commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth
change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or
topological change in its behaviour. Bifurcations occur in both continuous systems (described by ODEs, DDEs or
PDEs), and discrete systems (described by maps).

Bifurcation types

It is useful to divide bifurcations into two principal classes:

• Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria,
periodic orbits or other invariant sets as parameters cross through critical thresholds; and

• Global bifurcations, which often occur when larger invariant sets of the system 'collide' with each other, or with
equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed points).

Local bifurcations

A local bifurcation occurs when a parameter
change causes the stability of an equilibrium
(or fixed point) to change. In continuous
systems, this corresponds to the real part of
an eigenvalue of an equilibrium passing
through zero. In discrete systems (those
described by maps rather than ODEs), this
corresponds to a fixed point having a
Floquet multiplier with modulus equal to
one. In both cases, the equilibrium is
non-hyperbolic at the bifurcation point. The
topological changes in the phase portrait of
the system can be confined to arbitrarily
small neighbourhoods of the bifurcating
fixed points by moving the bifurcation
parameter close to the bifurcation point
(hence 'local').

More technically, consider the continuous
dynamical system described by the ODE

a. =-0.5

Phase portrait showing saddle-node bifurcation.

Period-halving bifurcations (L) leading to order, followed by period doubling
bifurcations (R) leading to chaos.

x = f(x, A) / :

Bifurcation theory 345

A local bifurcation occurs at (xq, Ao)if me Jacobian matrix df Xo A nas an eigenvalue with zero real part. If the
eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, but if the eigenvalue is non-zero but purely
imaginary, this is a Hopf bifurcation.
For discrete dynamical systems, consider the system

x n+l = J\ x rii -^ J ■

Then a local bifurcation occurs at (xq, Ao)if the matrix df Xo \ has an eigenvalue with modulus equal to one. If
the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps),
transcritical or pitchfork bifurcation. If the eigenvalue is equal to -1, it is a period-doubling (or flip) bifurcation, and
otherwise, it is a Hopf bifurcation.
Examples of local bifurcations include:

Saddle-node (fold) bifurcation

Transcritical bifurcation

Pitchfork bifurcation

Period-doubling (flip) bifurcation

Hopf bifurcation

Neimark (secondary Hopf) bifurcation

Global bifurcations

Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibria. This causes
changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as
is the case with local bifurcations. In fact, the changes in topology extend out to an arbitrarily large distance (hence
'global').

Examples of global bifurcations include:

• Homoclinic bifurcation in which a limit cycle collides with a saddle point.

• Heteroclinic bifurcation in which a limit cycle collides with two or more saddle points.

• Infinite-period bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle.

• Blue sky catastrophe in which a limit cycle collides with a nonhyperbolic cycle.

Global bifurcations can also involve more complicated sets such as chaotic attractors.

Codimension of a bifurcation

The codimension of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This
corresponds to the codimension of the parameter set for which the bifurcation occurs within the full space of
parameters. Saddle-node bifurcations and Hopf bifurcations are the only generic local bifurcations which are really
codimension-one (the others all having higher codimension). However, often transcritical and pitchfork bifurcations
are also often thought of as codimension-one, because the normal forms can be written with only one parameter.

An example of a well-studied codimension-two bifurcation is the Bogdanov— Takens bifurcation.

Applications in semiclassical and quantum physics

Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in
atomic systems, molecular systems, and resonant tunneling diodes. Bifurcation theory has also been

applied to the study of laser dynamics and a number of theoretical examples which are difficult to access
experimentally such as the kicked top and coupled quantum wells. The dominant reason for the link between
quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of
classical orbits becomes large, as Martin Gutzwiller points out in his classic work on quantum chaos. Many

Bifurcation theory 346

kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including
saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection
bifurcations, tangent bifurcations, and cusp bifurcations.

Notes

[I] Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006). Differential Equations. London: Thompson, pp. 96-1 1 1. ISBN 0495012653.

[2] Gao, J.; Delos, J. B. (1997). "Quantum manifestations of bifurcations of closed orbits in the photoabsorption spectra of atoms in electric

fields". Phys. Rev. A 56 (1): 356-364. Bibcode 1997PhRvA..56..356G. doi:10.1103/PhysRevA.56.356.
[3] Peters, A. D.; Jaffe, C; Delos, J. B. (1994). "Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model". Phys.

Rev. Lett. 73 (21): 2825-2828. Bibcode 1994PhRvL..73.2825P. doi:10.1103/PhysRevLett.73.2825. PMID 10057205.
[4] Courtney, M.; et al, H; Spellmeyer, N; Kleppner, D; Gao, J; Delos, JB (1995). "Closed Orbit Bifurcations in Continuum Stark Spectra".

Phys. Rev. Lett. 74 (9): 1538-1541. Bibcode 1995PhRvL..74.1538C. doi:10.1103/PhysRevLett.74.1538. PMID 10059054.
[5] Founargiotakis, M.; Farantos, S. C; Skokos, Ch.; Contopoulos, G. (1997). "Bifurcation diagrams of periodic orbits for unbound molecular

systems: FH2". Chemical Physics Letters 111 (5-6): 456-464. Bibcode 1997CPL...277..456F. doi: 10.1016/S0009-26 14(97)00931-7.
[6] Monteiro, T. S.; Saraga, D. S. (2001). "Quantum Wells in Tilted Fields: Semiclassical Amplitudes and Phase Coherence Times". Foundations

of Physics 31 (2): 355-370. doi:10.1023/A:1017546721313.
[7] Wieczorek, S.; Krauskopf, B.; Simpson, T. B.; Lenstra, D. (2005). "The dynamical complexity of optically injected semiconductor lasers".

Physics Reports 416 (1-2): 1-128. Bibcode 2005PhR...416....1W. doi:10.1016/j.physrep.2005.06.003.
[8] Stamatiou, G.; Ghikas, D. P. K. (2007). "Quantum entanglement dependence on bifurcations and scars in non-autonomous systems. The case

of quantum kicked top". Physics Letters A 368 (3-4): 206-214. arXiv:quant-ph/0702172. Bibcode 2007PhLA..368..206S.

doi:10.1016/j.physleta.2007.04.003.
[9] Galan, J.; Freire, E. (1999). "Chaos in a Mean Field Model of Coupled Quantum Wells; Bifurcations of Periodic Orbits in a Symmetric

Hamiltonian System". Reports on Mathematical Physics 44 (1-2): 87-94. Bibcode 1999RpMP...44...87G.

doi:10.1016/S0034-4877(99)80148-7.
[10] Kleppner, D.; Delos, J. B. (2001). "Beyond quantum mechanics: Insights from the work of Martin Gutzwiller". Foundations of Physics 31

(4): 593-612. doi:10.1023/A:1017512925106.

[II] Gutzwiller, Martin C. (1990). Chaos in Classical and Quantum Mechanics. New York: Springer- Verlag. ISBN 0387971734.

References

• Nonlinear dynamics (http://monet.physik.unibas.ch/~elmer/pendulum/nldyn.htm)

• Bifurcations and Two Dimensional Flows (http://www.egwald.ca/nonlineardynamics/bifurcations.php) by
Elmer G. Wiens

• Introduction to Bifurcation theory (http://prola.aps.org/abstract/RMP/v63/i4/p991_l) by John David
Crawford

• V. S. Afrajmovich, V. I. Arnold, et al, Bifurcation Theory And Catastrophe Theory, ISBN 3540653791

Rossler attractor

347

Rossler attractor

The Rossler attractor (pronounced

/rosier/) is the attractor for the Rossler

system, a system of three non-linear

ordinary differential equations. These

differential equations define a

continuous-time dynamical system that

exhibits chaotic dynamics associated with

the fractal properties of the attractor. Some

properties of the Rossler system can be

deduced via linear methods such as

eigenvectors, but the main features of the

system require non-linear methods such as

Poincare maps and bifurcation diagrams.

The original Rossler paper says the Rossler

attractor was intended to behave similarly to

the Lorenz attractor, but also be easier to

analyze qualitatively. An orbit within the

attractor follows an outward spiral close to

the £, y plane around an unstable fixed

point. Once the graph spirals out enough, a

second fixed point influences the graph,

causing a rise and twist in the z -dimension. In the time domain, it becomes apparent that although each variable is

oscillating within a fixed range of values, the oscillations are chaotic. This attractor has some similarities to the

Lorenz attractor, but is simpler and has only one manifold. Otto Rossler designed the Rossler attractor in 1976, but

the originally theoretical equations were later found to be useful in modeling equilibrium in chemical reactions. The

defining equations are:

Rossler attractor

348

Rossler attractor as a stereogram with (j — Q.2. b = 0.2

c=U

~dl = ~ V
dy

dt
dz

x + ay

b + z{x — c)

Rossler studied the chaotic attractor with a = 0.2> & = 0.2> an d c
}) = 0.1, and c = 14 nave been more commonly used since.

5.7, though properties of a = Q.l,

An analysis

Some of the Rossler attractor's elegance is due to two of its equations
being linear; setting ^ = 0> allows examination of the behavior on
the X, y plane

3*5 y plane of Rossler attractor with

a = 0.2, 6 = 0.2, c = 5.7

Rossler attractor 349

~dt
dy

dt V

dt X + aV

The stability in the X, y plane can then be found by calculating the eigenvalues of the Jacobian I , which

are (a =b V 'a 2 — 4)/2- From this, we can see that when < a < 2> the eigenvalues are complex and both have
a positive real component, making the origin unstable with an outwards spiral on the £, y plane. Now consider the
Z plane behavior within the context of this range for a . So long as x is smaller than c , the c term will keep the
orbit close to the £, y plane. As the orbit approaches x greater than c , the z -values begin to climb. As z climbs,
though, the —z in the equation for dx/dt stops the growth in x .

Fixed points

In order to find the fixed points, the three Rossler equations are set to zero and the ( x , V , Z ) coordinates of each
fixed point were determined by solving the resulting equations. This yields the general equations of each of the fixed
point coordinates:

:± \J(? - Aab

(c±\/c 2 -Aab"
y ~~{ 2a

c ± \Jc 2 — 4ab

z =

Which in turn can be used to show the actual fixed points for a given set of parameter values:
fc+ \/c 2 - Aab -c - \J(? - Aab c + \/c 2 - 4ab\

2 2a ' 2a

\/c 2 - Aab -c + \/c 2 - Aab c - \/c 2 - Aab''

\ 2 2a ' 2a )

As shown in the general plots of the Rossler Attractor above, one of these fixed points resides in the center of the
attractor loop and the other lies comparatively removed from the attractor.

Eigenvalues and eigenvectors

The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and
eigenvectors. Beginning with the Jacobian:

'0 -1 -1
1 a
K z x — Cj

the eigenvalues can be determined by solving the following cubic:

—A + A (a + x — c) + A(ac — ax — 1 — z) + x — c + az =

For the centrally located fixed point, Rossler's original parameter values of a=0.2, b=0.2, and c=5.7 yield eigenvalues
of:

Ai = 0.0971028 + 0.995786z
A 2 = 0.0971028 - 0.995786i
A 3 = -5.68718

Rossler attractor

350

Central Fixed Point Eigenvectors Examined

(Using Mathematica 7)

The magnitude of a negative eigenvalue characterizes the level of attraction along the corresponding eigenvector.
Similarly the magnitude of a positive eigenvalue characterizes the level of repulsion along the corresponding
eigenvector.

The eigenvectors corresponding to these eigenvalues are:

0.7073
v x = | -0.07278 - 0.7032z
0.0042 - 0.0007i

0.7073
v 2 = [ 0.07278 + 0.7032;
0.0042 + 0.0007i

0.1682
v 3 = [ -0.0286
0.9853

These eigenvectors have several

interesting implications. First, the two

eigenvalue/eigenvector pairs ( l>iand

"^2) are responsible for the steady

outward slide that occurs in the main

disk of the attractor. The last

eigenvalue/eigenvector pair is

attracting along an axis that runs

through the center of the manifold and

accounts for the z motion that occurs

within the attractor. This effect is

roughly demonstrated with the figure y

below. Examination of central fixed point eigenvectors: The blue line corresponds to the standard

Rossler attractor generated with (j — Q.2- b = 0.2' anc ' C = 5.7-

The figure examines the central fixed

point eigenvectors. The blue line

corresponds to the standard Rossler

attractor generated with a = 0.2- & = 0.2> anc ^ c = 5.7- The

red dot in the center of this attractor is FP\ ■

The red line intersecting that fixed

point is an illustration of the repulsing

plane generated by V\ and Vi . The green line

is an illustration of the attracting V3 . The

magenta line is generated by stepping

backwards through time from a point

on the attracting eigenvector which is slightly above FP\ — it illustrates the behavior of points that become

completely dominated by that vector. Note that the magenta line nearly touches the plane of the attractor before

being pulled upwards into the fixed point; this suggests that the general appearance and behavior of the Rossler

attractor is largely a product of the interaction between the attracting t>3and the repelling t>iand V% plane. Specifically it

implies that a sequence generated from the Rossler equations will begin to loop around FP\, start being pulled upwards

into the 1*3 vector, creating the upward arm of a curve

Rossler attractor with q = Q.2' h

c = 5.7'

0.2.

Rossler attractor 35 1

that bends slightly inward toward the vector before being pushed outward again as it is pulled back towards the
repelling plane.

For the outlier fixed point, Rossler's original parameter values of a = fj.2> b = 0.2> an ^ C = 5.7y ielc l
eigenvalues of:

Ai = -0.0000046 + 5.4280259z
A 2 = -0.0000046 - 5.4280259z
A 3 = 0.1929830

The eigenvectors corresponding to these eigenvalues are:

/0.0002422 + 0.1872055A
v x = 0.0344403 - 0.0013136i
\ 0.9817159 /

/0.0002422- 0.1872055 i\
v 2 = 0.0344403 + 0.0013136i
\ 0.9817159 /

/ 0.0049651 \
v 3 = -0.7075770
\ 0.7066188 /

Although these eigenvalues and eigenvectors exist in the Rossler attractor, their influence is confined to iterations of
the Rossler system whose initial conditions are in the general vicinity of this outlier fixed point. Except in those
cases where the initial conditions lie on the attracting plane generated by Ai a nd \ 2 , this influence effectively
involves pushing the resulting system towards the general Rossler attractor. As the resulting sequence approaches the
central fixed point and the attractor itself, the influence of this distant fixed point (and its eigenvectors) will wane.

Poincare map

The Poincare map is constructed by plotting the value of the function

every time it passes through a set plane in a specific direction. An

example would be plotting the y, z value every time it passes through

the x = 0pl ane where x is changing from negative to positive,

commonly done when studying the Lorenz attractor. In the case of the

Rossler attractor, the x = 0pl ane * s uninteresting, as the map always

crosses the x = 0pl ane at Z = O^ue to the nature of the Rossler

equations. In the x = 0.1pl a ne for a = 0.1, b = 0.L C = 14,

the Poincare map shows the upswing in z values as x increases, as is

to be expected due to the upswing and twist section of the Rossler plot.

The number of points in this specific Poincare plot is infinite, but when

a different c value is used, the number of points can vary. For

example, with a c value of 4, there is only one point on the Poincare map, because the function yields a periodic

orbit of period one, or if the c value is set to 12.8, there would be six points corresponding to a period six orbit.

■

»*

B 10 12 14 IE 1£ 20 22

Poincare map for Rossler attractor with

a = 0.1.6=0.1.c=14

Rossler attractor

352

Local Maxima inZ versusihe-projrimacsZrT

I ;

Mapping local maxima

In the original paper on the Lorenz Attractor, Edward Lorenz analyzed
the local maxima of z against the immediately preceding local
maxima. When visualized, the plot resembled the tent map, implying
that similar analysis can be used between the map and attractor. For the
Rossler attractor, when the Z n local maximum is plotted against the
next local z maximum, 2n+i, the resulting plot (shown here for
a = 0.2> & = 0.2> C=5.7)i s unimodal, resembling a skewed
Henon map. Knowing that the Rossler attractor can be used to create a
pseudo 1-d map, it then follows to use similar analysis methods. The
bifurcation diagram is specifically a useful analysis method.

Variation of parameters

Rossler attractor's behavior is largely a factor of the values of its constant parameters a , h , and c . In general,
varying each parameter has a comparable effect by causing the system to converge toward a periodic orbit, fixed
point, or escape towards infinity, however the specific ranges and behaviors induced vary substantially for each
parameter. Periodic orbits, or "unit cycles," of the Rossler system are defined by the number of loops around the
central point that occur before the loops series begins to repeat itself.

Bifurcation diagrams are a common tool for analyzing the behavior of dynamical systems, of which the Rossler
attractor is one. They are created by running the equations of the system, holding all but one of the variables constant
and varying the last one. Then, a graph. is plotted of the points that a particular value for the changed variable visits
after transient factors have been neutralised. Chaotic regions are indicated by filled-in regions of the plot.

(

; v ...

5 10 IS

Z n vs. Z n+ i

Varying a

Here, 5 is fixed at 0.2, cis fixed at 5.7 and a changes. Numerical examination of the attractor's behavior over
changing a suggests it has a disproportional influence over the attractor's behavior. The results of the analysis are:

• a < : Converges to the centrally located fixed point

• a = 0.1 ; Unit cycle of period 1

• a = 0.2 ; Standard parameter value selected by Rossler, chaotic

• a = 0.3 ; Chaotic attractor, significantly more Mobius strip-like (folding over itself).

• a = 0.35 : Similar to .3, but increasingly chaotic

• a = 0.38 ; Similar to .35, but increasingly chaotic.

Rossler attractor

353

Varying b

Here, a is fixed at 0.2, cis fixed at
5.7 and 5 changes. As shown in the
accompanying diagram, as

approaches the attractor approaches
infinity (note the upswing for very
small values of 5 ■ Comparative to the
other parameters, varying 5 generates
a greater range when period-3 and
period-6 orbits will occur. In contrast
to a and c , higher values of £
converge to period- 1, not to a chaotic
state.

BRmUnnwmtoM^rMMCMtwMN

b "

1 m.

1 Xt^P ' - J__> _____

-1 :k 'y — — " ~~~ — :

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Bifurcation diagram for the Rossler attractor for varying fa

Varying c

Here, a = 6 = 0.1 an d C changes.
The bifurcation diagram reveals that
low values of c are periodic, but
quickly become chaotic as c increases.
This pattern repeats itself as c
increases — there are sections of
periodicity interspersed with periods of
chaos, and the trend is towards
higher-period orbits as c increases.
For example, the period one orbit only
appears for values of c around 4 and is
never found again in the bifurcation
diagram. The same phenomena is seen
with period three; until q = \%
period three orbits can be found, but
thereafter, they do not appear.

A graphical illustration of the changing

attractor over a range of c values

illustrates the general behavior seen for all of these parameter analyses — the frequent transitions between periodicity

and aperiodicity.

Bifurcation diagram

______T

? fc JflfHWHPIM-'J.JM

S 10 1"! 20 25 30 35 40

Bifurcation diagram for the Rossler attractor for varying C

Rossler attractor

354

^ 3 ■

* 1

1 1

-3

/ /

) 4

1Q 1!

(a) c=A, period 1

(b) c=6, period 2

(g) c=12.6, period 6

-a o % io

(h}c=13, chaotic

"^tt -M 3 ^ a" ifl is

-14 -IB -i D * H 15 »

(f) c=12

period 3

:■;■

— U^ ^ ..

Mj/y, &r~^~

^-5^^^.

mtm

J' 'till fwjift

D 10 3J X

(i)c=18, chaotic

The above set of images illustrates the variations in the post-transient Rossler system as c is varied over a range of
values. These images were generated with q = 5 = .1-

c = 4> period- 1 orbit.

q = Q, period-2 orbit.

c = 8 5' period-4 orbit.

q = 3.7' period-8 orbit.

c = 9' sparse chaotic attractor.

C = 12' period-3 orbit.

C = 12.6. period-6 orbit.

q = 13' sparse chaotic attractor.

c = 18' filled-in chaotic attractor.

Rossler attractor 355

Links to other topics

The banding evident in the Rossler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the
half-twist in the Rossler attractor makes it similar to a Mobius strip.

References

• E. N. Lorenz (1963). "Deterministic nonperiodic flow". /. Atmos. Sci. 20 (2): 130—141.
doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. ISSN 1520-0469.

• O. E. Rossler (1976). "An Equation for Continuous Chaos". Physics Letters 57A (5): 397-398.

• O. E. Rossler (1979). "An Equation for Hyperchaos". Physics Letters 71A (2,3): 155-157.

• Steven H. Strogatz (1994). Nonlinear Dynamics and Chaos. Perseus publishing.

External links

• Fla

• [2]

Flash Animation using PovRay

Lorenz and Rossler attractors — Java animation

3D Attractors: Mac program to visualize and explore the Rossler and Lorenz attractors in 3 dimensions

Rossler attractor in Scholarpedia

References

[1] http://lagrange.physics.drexel.edu/flash/rossray

[2] http://www.soe.ucsc.edu/classes/ams214/Winter09/foundingpapers/Rosslerl976.pdf

[3] http://scholarpedia.org/article/Rossler_attractor

Synchronizing Chaos

Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic oscillators are coupled, or
when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly effect, which causes the
exponential divergence of the trajectories of two identical chaotic system started with nearly the same initial
conditions, having two chaotic system evolving in synchrony might appear quite surprising. However,
synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and
reasonably well understood theoretically.

It has been found that chaos synchronization is quite a rich phenomenon that may present a variety of forms. When
two chaotic oscillators are considered, these include:

• Identical synchronization. This is a straightforward form of synchronization that may occur when two identical
chaotic oscillators are mutually coupled, or when one of them drives the other. If (x ,x „...,x ) and (x' , x' ,...,x' )
denote the set of dynamical variables that describe the state of the first and second oscillator, respectively, it is
said that identical synchronization occurs when there is a set of initial conditions [x (0), x (0),...,x (0)], [x' (0),
x' (0),...,x' (0)] such that, denoting the time by t, lx'.(t)-x.((t)l— >0, for i=l,2,...,n, when t— >°°. That means that for
time large enough the dynamics of the two oscillators verifies x'.(t)=x.(t), for i=l,2,...,n, in a good approximation.
This is called the synchronized state in the sense of identical synchronization.

• Generalized synchronization. This type of synchronization occurs mainly when the coupled chaotic oscillators
are different, although it has also been reported between identical oscillators. Given the dynamical variables
(x,,x„„...,x ) and (y,,y„„...,y ) that determine the state of the oscillators, generalized synchronization occurs

1 2 n J \ J 2 J m ° J

when there is a functional, O, such that, after a transitory evolution from appropriate initial conditions, it is [y.(t),

Synchronizing Chaos 356

y (t),...,y (t)]=0[x (t), x (t),...,x (t)]. This means that the dynamical state of one of the oscillators is completely
determined by the state of the other. When the oscillators are mutually coupled this functional has to be invertible,
if there is a drive-response configuration the drive determines the evolution of the response, and O does not need
to be invertible. Identical synchronization is the particular case of generalized synchronization when O is the
identity.

• Phase synchronization. This form of synchronization, which occurs when the oscillators coupled are not
identical, is partial in the sense that, in the synchronized state, the amplitudes of the oscillator remain
unsynchronized, and only their phases evolve in synchrony. Observation of phase synchronization requires a
previous definition of the phase of a chaotic oscillator. In many practical cases, it is possible to find a plane in
phase space in which the projection of the trajectories of the oscillator follows a rotation around a well-defined
center. If this is the case, the phase is defined by the angle, <p(t), described by the segment joining the center of
rotation and the projection of the trajectory point onto the plane. In other cases it is still possible to define a phase
by means of techniques provided by the theory of signal processing, such as the Hilbert transform. In any case, if
cp (t) and cp.(t) denote the phases of the two coupled oscillators, synchronization of the phase is given by the
relation nep (t)=mq) (t) with m and n whole numbers.

• Anticipated and lag synchronization. In these cases the synchronized state is characterized by a time interval x

such that the dynamical variables of the oscillators, (x ,x ,,...,x ) and (x' , x' ,...,x' ), are related by x'.(t)=x.(t+x);
this means that the dynamics of one of the oscillators follows, or anticipates, the dynamics of the other.
Anticipated synchronization may occur between chaotic oscillators whose dynamics is described by delay
differential equations, coupled in a drive-response configuration. In this case, the response anticipates the
dynamics of the drive. Lag synchronization may occur when the strength of the coupling between
phase-synchronized oscillators is increased.

• Amplitude envelope synchronization. This is a mild form of synchronization that may appear between two
weakly coupled chaotic oscillators. In this case, there is no correlation between phases nor amplitudes; instead,
the oscillations of the two systems develop a periodic envelope that has the same frequency in the two systems.
This has the same order of magnitude than the difference between the average frequencies of oscillation of the
two chaotic oscillator. Often, amplitude envelope synchronization precedes phase synchronization in the sense
that when the strength of the coupling between two amplitude envelope synchronized oscillators is increased,
phase synchronization develops.

All these forms of synchronization share the property of asymptotic stability. This means that once the synchronized
state has been reached, the effect of a small perturbation that destroys synchronization is rapidly damped, and
synchronization is recovered again. Mathematically, asymptotic stability is characterized by a positive Lyapunov
exponent of the system composed of the two oscillators, which becomes negative when chaotic synchronization is
achieved.

Some chaotic systems allow even stronger control of chaos. Both synchronization of chaos and control of chaos
constitute parts of Cybernetical Physics.

Synchronizing Chaos

357

Books

• Pikovsky, A.; Rosemblum, M.; Kurths, J. (2001). Synchronization: A Universal Concept in Nonlinear Sciences.
Cambridge University Press. ISBN 0-521-53352-X.

• Gonzalez-Miranda, J. M. (2004). Synchronization and Control of Chaos. An introduction for scientists and
engineers. Imperial College Press. ISBN 1-86094-488-4.

• Fradkov A.L.. Cybernetical physics: from control of chaos to quantum control. Springer-Verlag, 2007,
(Preliminary Russian version: St. Petersburg, Nauka, 2003)..

The Possibility of Quantum Chaos ?

Quantum chaos is a branch of physics which studies
how chaotic classical dynamical systems can be
described in terms of quantum theory. The primary
question that quantum chaos seeks to answer is, "What
is the relationship between quantum mechanics and
classical chaos?" The correspondence principle states
that classical mechanics is the classical limit of
quantum mechanics. If this is true, then there must be
quantum mechanisms underlying classical chaos;
although this may not be a fruitful way of examining
classical chaos. If quantum mechanics does not
demonstrate an exponential sensitivity to initial
conditions, how can exponential sensitivity to initial
conditions arise in classical chaos, which must be the
correspondence principle limit of quantum mechanics?
In seeking to address the basic question of
quantum chaos, several approaches have been
employed:

What is Quantum Chaos?

Classical
World

Quantum
World

■ Correspondence Principle

Wavepacket Dynamics

Dephasing

Energy-level Statistics

Quantization Schemes ■

WKB/EBKM Tori Quantization

Periodic-Orbit Theory

Closed-Orbit Theory

Quantum chaos is the field of physics attempting to build a bridge

between the theories of quantum mechanics and classical dynamics.

The figure shows the main ideas running in each direction.

1 . Development of methods for solving quantum
problems where the perturbation cannot be
considered small in perturbation theory and where
quantum numbers are large.

2. Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same
Hamiltonian (system).

3. Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system
with quantum features.

4. Direct application of the correspondence principle.

The Possibility of Quantum Chaos '

358

History

strength

During the first half of the twentieth
century, chaotic behavior in mechanics was
recognized (as in the three-body problem in
celestial mechanics), but not

well-understood. The foundations of modern
quantum mechanics were laid in that period,
essentially leaving aside the issue of the
quantum-classical correspondence in
systems whose classical limit exhibit chaos.

Experimental recurrence spectra of lithium in an electric field showing birth of

[3]
quantum recurrences corresponding to bifurcations of classical orbits.

Approaches

-ri

4-1
til

Questions related to the correspondence
principle arise in many different branches of
physics, ranging from nuclear to atomic,
molecular and solid-state physics, and even
to acoustics, microwaves and optics.
Important observations often associated with
classically chaotic quantum systems are
spectral level repulsion, dynamical
localization in time evolution (e.g.
ionization rates of atoms), and enhanced
stationary wave intensities in regions of
space where classical dynamics exhibits
only unstable trajectories (as in scattering).

In the semiclassical approach of quantum

chaos, phenomena are identified in

spectroscopy by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with

classical orbits. Other phenomena show up in the time evolution of a quantum system, or in its response to various

types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable

and exhibit irregular amplitude distributions.

Quantum chaos typically deals with systems whose properties need to be calculated using either numerical
techniques or approximation schemes (see e.g. Dyson series). Simple and exact solutions are precluded by the fact

scaled action

Comparison of experimental and theoretical recurrence spectra of lithium in an
electric field at a scaled energy of g = — 3.0-

The Possibility of Quantum Chaos '

359

that the system's constituents either influence each other in a complex way, or depend on temporally varying external
forces.

Quantum Mechanics in Non-Perturbative Regimes

For conservative systems, the goal of
quantum mechanics in non-perturbative
regimes is to find the eigenvalues and
eigenvectors of a Hamiltonian of the form

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
F [V/cm]

Computed regular (non-chaotic) Rydberg atom energy level spectra of hydrogen in
an electric field near n=15. Note that energy levels can cross due to underlying
symmetries of dynamical motion.

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
F [V/cm]

Computed chaotic Rydberg atom energy level spectra of lithium in an electric field

near n=15. Note that energy levels cannot cross due to the ionic core (and resulting

[4]
quantum defect) breaking symmetries of dynamical motion.

H = H s + eH ns ,
where iJ s is separable in some coordinate system, H ns i& non-separable in the coordinate system in which // s is
separated, and e is a parameter which cannot be considered small. Physicists have historically approached problems
of this nature by trying to find the coordinate system in which the non-separable Hamiltonian is smallest and then
treating the non-separable Hamiltonian as a perturbation.

Finding constants of motion so that this separation can be performed can be a difficult (sometimes impossible)
analytical task. Solving the classical problem can give valuable insight into solving the quantum problem. If there are

The Possibility of Quantum Chaos ? 360

regular classical solutions of the same Hamiltonian, then there are (at least) approximate constants of motion, and by
solving the classical problem, we gain clues how to find them.

Other approaches have been developed in recent years. One is to express the Hamiltonian in different coordinate
systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region.
Wavefunctions are obtained in these regions, and eigenvalues are obtained by matching boundary conditions.

Another approach is numerical matrix diagonalization. If the Hamiltonian matrix is computed in any complete basis,
eigenvalues and eigenvectors are obtained by diagonalizing the matrix. However, all complete basis sets are infinite,
and we need to truncate the basis and still obtain accurate results. These techniques boil down to choosing a
truncated basis from which accurate wavefunctions can be constructed. The computational time required to
diagonalize a matrix scales as jy 3 , where _/\Tis the dimension of the matrix, so it is important to choose the
smallest basis possible from which the relevant wavefunctions can be constructed. It is also convenient to choose a
basis in which the matrix is sparse and/or the matrix elements are given by simple algebraic expressions because
computing matrix elements can also be a computational burden.

A given Hamiltonian shares the same constants of motion for both classical and quantum dynamics. Quantum
systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity
conservation from reflection symmetry). However, if we merely find quantum solutions of a Hamiltonian which is
not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned
little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of
answering the question of quantum chaos.

The Possibility of Quantum Chaos '

361

Correlating Statistical Descriptions of Quantum Mechanics with Classical
Behavior

Statistical measures of quantum chaos were born out of
a desire to quantify spectral features of complex
systems. Random matrix theory was developed in an
attempt to characterize spectra of complex nuclei. The
remarkable result is that the statistical properties of
many systems with unknown Hamiltonians can be
predicted using random matrices of the proper
symmetry class. Furthermore, random matrix theory
also correctly predicts statistical properties of the
eigenvalues of many chaotic systems with known
Hamiltonians. This makes it useful as a tool for
characterizing spectra which require large numerical
efforts to compute.

A number of statistical measures are available for
quantifying spectral features in a simple way. It is of
great interest whether or not there are universal
statistical behaviors of classically chaotic systems. The
statistical tests mentioned here are universal, at least to
systems with few degrees of freedom (Berry and Tabor
have put forward strong arguments for a Poisson
distribution in the case of regular motion and Heusler et
al. present a semiclassical explanation of the
so-called Bohigas-Giannoni-Schmit conjecture which
asserts universality of spectral fluctuations in chaotic
dynamics). The nearest-neighbor distribution (NND) of
energy levels is relatively simple to interpret and it has
been widely used to describe quantum chaos.

Nearest neighbor distribution for Rydberg atom energy level spectra

in an electric field as quantum defect is increased from 0.04 (a) to

0.32 (h). The system becomes more chaotic as dynamical symmetries

are broken by increasing the quantum defect; consequently, the

distribution evolves from nearly a Poisson distribution (a) to a

Wigner distribution (h).

Qualitative observations of level repulsions can be

quantified and related to the classical dynamics using the NND, which is believed to be an important signature of
classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a Poisson
distribution of energy levels:

P(s) = e~ s .
In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of
random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a
number of chaotic systems have been shown to be in good agreement with the predictions of the Gaussian
orthogonal ensemble (GOE) of random matrices, and it has been suggested that this phenomenon is generic for all
chaotic systems with this symmetry. If the normalized spacing between two energy levels is s , the normalized
distribution of spacings is well approximated by

■K

P(s) = -se

s 2 /4

which is the Wigner distribution.

Many Hamiltonian systems which are classically integrable (non-chaotic) have been found to have quantum
solutions that yield nearest neighbor distributions which follow the Poisson distributions. Similarly, many systems

The Possibility of Quantum Chaos '

362

which exhibit classical chaos have been found with quantum solutions yielding a Wigner distribution, thus
supporting the ideas above. One notable exception is diamagnetic lithium which, though exhibiting classical chaos,
demonstrates Wigner (chaotic) statistics for the even-parity energy levels and nearly Poisson (regular) statistics for
the odd-parity energy level distribution.

Semiclassical Methods

Periodic Orbit Theory

Periodic -orbit theory gives a recipe for
computing spectra from the periodic
orbits of a system. In contrast to the
Einstein-Brillouin-Keller method of
action quantization, which applies only
to integrable or near-integrable
systems and computes individual
eigenvalues from each trajectory,
periodic-orbit theory is applicable to
both integrable and non-integrable
systems and asserts that each periodic
orbit produces a sinusoidal fluctuation
in the density of states.

/7SJ30

v 2

o s

0.5 1 1.5 2 2.5 3

Even parity recurrence spectrum (Fourier transform of the density of
states) of diamagnetic hydrogen showing peaks corresponding to

periodic orbits of the classical system. Spectrum is at a scaled energy
of -0.6. Peaks labeled R and V are repetitions of the closed orbit

perpendicular and parallel to the field, respectively. Peaks labeled O

correspond to the near circular periodic orbit that goes around the

nucleus.

\\ 1 1 1 1 1

1 40

\
1

1 30

- \ —

a>
o

§ 20

3
O

a>
* 10

1 1 1 1 1 1

2 4 6 8 10 12

Relative recurrence amplitudes of even and odd recurrences of the near circular orbit.

Diamonds and plus signs are for odd and even quarter periods, respectively. Solid line is

A/cosh(nX/8). Dashed line is A/sinh(nX/8) where A = 14.75 and X = 1.18.

The principal result of this development is an expression for the density of states which is the trace of the
semiclassical Green's function and is given by the Gutzwiller trace formula:

The Possibility of Quantum Chaos ? 363

OC -1

gJE) = YT k Y — S nS "- a ^ ir ' 2 \

The index fc distinguishes the primitive periodic orbits: the shortest period orbits of a given set of initial conditions.

Tj. is the period of the primitive periodic orbit and S^ is its classical action. Each primitive orbit retraces itself,

leading to a new orbit with action nS^ and a period which is an integral multiple n of the primitive period. Hence,

every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the

intermediate sum over the indices n ■ CX-nk is the orbit's Maslov index. The amplitude factor, 1 / sinh. [Xnkl'^) >

represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit

diverge exponentially in time from the periodic orbit. The quantity Xnk characterizes the instability of the orbit. A

stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. For stable orbits,

sinh. (Xnfe/2) becomes sin (Xn.fc/2) » where Xnk is the winding number of the periodic orbit. Xnk = 27rm ,

where m is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a

difficulty because sin (Xnfc/2) = Oat a classical bifurcation. This causes that orbit's contribution to the energy

density to diverge. This also occurs in the context of photo-absorption spectrum.

Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This

presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a

function of action. 2) There are an infinite number of periodic orbits, and the convergence properties of

periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular

systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to

roundoff errors and details of the numerical integration.

Gutz wilier applied the trace formula to approach the anisotropic Kepler problem (a single particle in a \/r potential
with an anisotropic mass tensor) semiclassically. He found agreement with quantum computations for low lying (up
to 77, = 6) states for small anisotropics by using only a small set of easily computed periodic orbits, but the
agreement was poor for large anisotropics.

The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each
periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties
surrounding long-period orbits to try and find the density of states (energy levels), one can use standard quantum
mechanical perturbation theory to compute eigenvalues (energy levels) and use the Fourier transform to look for the
periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then
amounts to finding the orbits which correspond to peaks in the Fourier transform.

The Possibility of Quantum Chaos

364

Closed Orbit Theory

Closed-orbit theory was developed by J.B. Delos, M.L.
Du, J. Gao, and J. Shaw. It is similar to periodic-orbit
theory, except that closed-orbit theory is applicable
only to atomic and molecular spectra and yields the
oscillator strength density (observable photo-absorption
spectrum) from a specified initial state whereas
periodic-orbit theory yields the density of states.

Only orbits that begin and end at the nucleus are
important in closed-orbit theory. Physically, these are
associated with the outgoing waves that are generated
when a tightly bound electron is excited to a high-lying
state. For Rydberg atoms and molecules, every orbit
which is closed at the nucleus is also a periodic orbit
whose period is equal to either the closure time or twice
the closure time.

Experimental recurrence spectrum (circles) is compared with the

results of the closed orbit theory of John Delos and Jing Gao for

lithium Rydberg atoms in an electric field. The peaks labeled 1-5 are

repetitions of the electron orbit parallel to the field going from the

nucleus to the classical turning point in the uphill direction.

According to closed-orbit theory, the average oscillator strength density at constant e is given by a smooth
background plus an oscillatory sum of the form f{w} = y^ /^ ^nk sin{2TrnwSj i: — (j) n k).

k tj=1

<^„£ is a phase that depends on the Maslov index and other details of the orbits. JJ 1 , is the recurrence amplitude of

a closed orbit for a given initial state (labeled j ). It contains information about the stability of the orbit, its initial
and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy
Coulomb wave. For scaling systems such as Rydberg atoms in strong fields, the Fourier transform of an oscillator
strength spectrum computed at fixed e as a function of w is called a recurrence spectrum, because it gives peaks
which correspond to the scaled action of closed orbits and whose heights correspond to JJ 1 , .

Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen,
hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the jj— ion in
crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.

Recent directions in quantum chaos

The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the
study of eigenfunctions (Quantum ergodicity, scars) of various chaotic Hamiltonian H(x,p; R) ■

Further studies concern the parametric ( JJ ) dependence of the Hamiltonian, as reflected in e.g. the statistics of
avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is
vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility
issues etc. Special place is reserved to the study of the dynamics of quantized maps: The Standard map and The
Kicked Rotator are considered to be prototype problems.

ron

Recent works are also focused in the study of driven chaotic systems, where the Hamiltonian H(x,p; i?(t))is
time dependent, in particular in the adiabatic and in the linear response regimes.

The Possibility of Quantum Chaos ? 365

References

[1] Quantum Signatures of Chaos, Fritz Haake, Edition: 2, Springer, 2001, ISBN 3540677232, 9783540677239.

[2] Michael Berry, Quantum Chaology, ppl04-5 of Quantum: a guide for the perplexed by Jim Al-Khalili (Weidenfeld and Nicolson 2003), http:/

/www. physics. bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf.
[3] Closed Orbit Bifurcations in Continuum Stark Spectra, M Courtney, H Jiao, N Spellmeyer, D Kleppner, J Gao, JB Delos, Phys Rev Let 27,

1538(1995).
[4] Classical, semiclassical, and quantum dynamics of lithium in an electric field, M Courtney, N Spellmeyer, H Jiao, D Kleppner, Phys Rev A

51,3604(1995).
[5] M.V. Berry and M. Tabor, Proc. Roy. Soc. London A 356, 375, 1977

[6] Heusler, S., S. Muller, A. Altland, P. Braun, and F. Haake, 2007, Phys. Rev. Lett. 98, 044103
[7] Courtney, M and Kleppner, D (http://pra.aps.org/abstract/PRA/v53/il/pl78_l), Core-induced chaos in diamagnetic lithium, PRA 53,

178, 1996
[8] Driven chaotic mesoscopic systems, dissipation and decoherence, in Proceedings of the 38th Karpacz Winter School of Theoretical Physics,

Edited by P. Garbaczewski and R. Olkiewicz (Springer, 2002). http://arxiv.org/abs/quant-ph/0403061

• Martin C. Gutzwiller (1971). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical
Physics 12: 343. doi: 10. 1063/1. 1665596.

• Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (1990) Springer- Verlag, New York
ISBN=0-387-97 173-4.

• Stockmann Hans-Jiirgen, Quantum Chaos: An Introduction, (1999) Cambridge University Press
ISBN=0-521-59284-4.

• Eugene Paul Wigner; Dirac, P. A. M. (1951). "On the statistical distribution of the widths and spacings of nuclear
resonance levels". Proc. Cambr. Philos. Soc. 47: 790. doi: 10. 1017/S0305004 100027237.

• Fritz Haake, Quantum Signatures of Chaos 2nd ed., (2001) Springer- Verlag, New York ISBN=3-540-67723-2.

• Quantum chaos on arxiv.org (http://xstructure.inr.ac. ru/x-bin/theme3.py?level=2&indexl=142714)

• Karl-Fredrik Berggren and Sven Aberg, "Quantum Chaos Y2K Proceedings of Nobel Symposium 1 16" (2001)
ISBN 978-9810247119

External links

• Quantum Chaos (http://www. sciam.com/article. cfm?id=quantum-chaos-subatomic-worlds) by Martin
Gutzwiller (1992, Scientific American)

• What is... Quantum Chaos (http://www.ams.org/notices/200801/tx080100032p.pdf) by Ze'ev Rudnick
(January 2008, Notices of the American Mathematical Society)

• Brian Hayes, "The Spectrum of Riemannium"; American Scientist (http://www.americanscientist.org/template/
AssetDetail/assetid/21879/page/l;jsessionid=aaa-ZYP5NrRxh8). Discusses relation to the Riemann zeta
function.

• Eigenfunctions in chaotic quantum systems (http://nbn-resolving.de/
urn:nbn:de:bsz:14-ds-1213275874643-50420) by Arnd Backer.

• Quantum Chaos at Scholarpedia (http://www.scholarpedia.Org/article/Category:Quantum_Chaos)

Fractals and Fractional Dimensions

366

Fractals and Fractional Dimensions

A fractal is "a rough or fragmented

geometric shape that can be split into

parts, each of which is (at least

approximately) a reduced-size copy of

the whole," a property called

self-similarity. Roots of the idea of

fractals go back to the 17th century,

while mathematically rigorous

treatment of fractals can be traced back

to functions studied by Karl

Weierstrass, Georg Cantor and Felix

Hausdorff a century later in studying

functions that were continuous but not

differentiable; however, the term

fractal was coined by Benoit

Mandelbrot in 1975 and was derived

from the Latin fractus meaning

"broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback

based on recursion. There are several examples of fractals, which are defined as portraying exact self-similarity,

quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in

nature, which has led to their inclusion in artwork. They are useful in medicine, soil mechanics, seismology, and

technical analysis.

The Mandelbrot set is a famous example of a fractal

Characteristics

A fractal often has the following features:

• It has a fine structure at arbitrarily small scales.

• It is too irregular to be easily described in traditional
Euclidean geometric language.

• It is self-similar (at least approximately or
stochastically).

• It has a Hausdorff dimension which is greater than its
topological dimension (although this requirement is not
met by space-filling curves such as the Hilbert
curve).

• It has a simple and recursive definition.

Because they appear similar at all levels of magnification,
fractals are often considered to be infinitely complex (in
informal terms). Natural objects that are approximated by
fractals to a degree include clouds, mountain ranges,
lightning bolts, coastlines, snow flakes, various vegetables
(cauliflower and broccoli), and animal coloration patterns.

Frost crystals formed naturally on cold glass illustrate fractal
process development in a purely physical system

Fractals and Fractional Dimensions

367

However, not all self-similar objects are fractals — for example, the real line (a straight Euclidean line) is formally
self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in
Euclidean terms.

Images of fractals can be created using fractal-generating software. Images produced by such software are normally
referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom
into a region of the fractal that does not exhibit any fractal properties. Also, these may include calculation or display
artifacts which are not characteristics of true fractals.

History

The mathematics behind fractals began to take shape in the 17th
century when a mathematician and philosopher Gottfried Leibniz
considered recursive self-similarity (although he made the mistake
of thinking that only the straight line was self-similar in this
sense).

It was not until 1872 that a function appeared whose graph would
today be considered fractal, when Karl Weierstrass gave an
example of a function with the non-intuitive property of being
everywhere continuous but nowhere differentiable. In 1904, Helge
von Koch, dissatisfied with Weierstrass's abstract and analytic
definition, gave a more geometric definition of a similar function,
which is now called the Koch curve. Waclaw Sierpihski
constructed his triangle in 1915 and, one year later, his carpet. The
idea of self-similar curves was taken further by Paul Pierre Levy,
who, in his 1938 paper Plane or Space Curves and Surfaces
Consisting of Parts Similar to the Whole described a new fractal
curve, the Levy C curve. Georg Cantor also gave examples of
subsets of the real line with unusual properties — these Cantor sets are also now recognized as fractals.

Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincare,
Felix Klein, Pierre Fatou and Gaston Julia. Without the aid of modern computer graphics, however, they lacked the
means to visualize the beauty of many of the objects that they had discovered.

In the 1960s, Benoit Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension} which built on earlier work by Lewis Fry
Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff— Besicovitch
dimension is greater than its topological dimension. He illustrated this mathematical definition with striking
computer-constructed visualizations. These images captured the popular imagination; many of them were based on
recursion, leading to the popular meaning of the term "fractal".

A Koch snowflake, which begins with an equilateral

triangle and then replaces the middle third of every line

segment with a pair of line segments that form an

equilateral "bump."

Fractals and Fractional Dimensions

368

Examples

A class of examples is given by the Cantor sets,
Sierpinski triangle and carpet, Menger sponge,
dragon curve, space-filling curve, and Koch
curve. Additional examples of fractals include the
Lyapunov fractal and the limit sets of Kleinian
groups. Fractals can be deterministic (all the
above) or stochastic (that is, non-deterministic).
For example, the trajectories of the Brownian
motion in the plane have a Hausdorff dimension
of 2.

Chaotic dynamical systems are sometimes
associated with fractals. Objects in the phase
space of a dynamical system can be fractals (see

attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is
the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension
of 2 — but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2
(while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is
the Julia set.

Generation

A Julia set, a fractal related to the Mandelbrot set

The whole Mandelbrot set

Mandelbrot zoomed 6x

Mandelbrot Zoomed lOOx

Fractals and Fractional Dimensions 369

Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set

Four common techniques for generating fractals are:

• Escape-time fractals — (also known as "orbits" fractals) These are defined by a formula or recurrence relation at
each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the
Burning Ship fractal, the Nova fractal and the Lyapunov fractal. The 2d vector fields that are generated by one or
two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed
through this field repeatedly.

• Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet,
Sierpinski gasket, Peano curve, Koch snowflake, Harter-Highway dragon curve, T-Square, Menger sponge, are
some examples of such fractals.

• Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the
Brownian motion, Levy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or
dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.

• Strange attractors — Generated by iteration of a map or the solution of a system of initial-value differential
equations that exhibit chaos.

Classification

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in
fractals:

• Exact self-similarity — This is the strongest type of self-similarity; the fractal appears identical at different scales.
Fractals defined by iterated function systems often display exact self-similarity. For example, the Sierpinski
triangle and Koch snowflake exhibit exact self-similarity.

• Quasi-self-similarity — This is a looser form of self-similarity; the fractal appears approximately (but not exactly)
identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and
degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly
self-similar. The Mandelbrot set is quasi-self-similar, as the satellites are approximations of the entire set, but not
exact copies.

• Statistical self-similarity — This is the weakest type of self-similarity; the fractal has numerical or statistical
measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of
statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.)
Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor
quasi-self-similar. The coastline of Britain is another example; one cannot expect to find microscopic Britains
(even distorted ones) by looking at a small section of the coast with a magnifying glass.

Possessing self-similarity is not the sole criterion for an object to be termed a fractal. Examples of self-similar
objects that are not fractals include the logarithmic spiral and straight lines, which do contain copies of themselves at
increasingly small scales. These do not qualify, since they have the same Hausdorff dimension as topological
dimension.

Fractals and Fractional Dimensions

370

In nature

Approximate fractals are easily found in nature. These
objects display self-similar structure over an extended,

but finite, scale range. Examples include clouds, river

rxi
networks, fault lines, mountain ranges, craters, snow

flakes, crystals, lightning, cauliflower or broccoli,

and systems of blood vessels and pulmonary vessels,

and ocean waves . DNA and heartbeat can be

analyzed as fractals. Even coastlines may be loosely

considered fractal in nature.

A fractal that models the surface of a mountain (animation)

Trees and ferns are fractal in nature and can be

modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples — a branch
from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The
connection between fractals and leaves is currently being used to determine how much carbon is contained in

trees

[13]

In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance" — the same

1141
electromagnetic properties no matter what the frequency — from Maxwell's equations (see fractal antenna).

In creative works

Fractal patterns have been found in the paintings of
American artist Jackson Pollock. While Pollock's
paintings appear to be composed of chaotic dripping
and splattering, computer analysis has found fractal
patterns in his work.

Decalcomania, a technique used by artists such as Max
Ernst, can produce fractal-like patterns. It involves
pressing paint between two surfaces and pulling them
apart.

Cyberneticist Ron Eglash has suggested that fractal-like

structures are prevalent in African art and architecture.

Circular houses appear in circles of circles, rectangular

houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture,

and even cornrow hairstyles.

In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted that the structure of the first draft of
Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (aka

A fractal created using the program Apophysis

Sierpinski gasket) but that the edited novel is "more like a lopsided Sierpinsky Gasket"

[19]

Fractals and Fractional Dimensions

371

Applications

As described above, random fractals can be used to
describe many highly irregular real-world objects.
Other applications of fractals include:

Classification of histopathology slides in medicine

Fractal landscape or Coastline complexity

Enzyme/enzymology (Michaelis-Menten kinetics)

Generation of new music

Signal and image compression

Creation of digital photographic enlargements

Seismology

Fractal in soil mechanics

Computer and video game design, especially

computer graphics for organic environments and as

part of procedural generation

Fractography and fracture mechanics

Fractal antennas — Small size antennas using fractal

shapes

Small angle scattering theory of fractally rough

systems

T-shirts and other fashion

Generation of patterns for camouflage, such as

MARPAT

Digital sundial

Technical analysis of price series (see Elliott wave

principle)

Fractals in networks

A fractal is formed when pulling apart two glue-covered acrylic
sheets

High voltage breakdown within a 4" block of acrylic creates a fractal
Lichtenberg figure

References

Nots

[1] Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.. ISBN 0-7167-1186-9.

[2] Briggs, John (1992). FractalsiThe Patterns of Chaos. London : Thames and Hudson, 1992.. p. 148. ISBN 0500276935, 0500276935.

[3] Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.. xxv.

ISBN 0-470-84862-6.
[4] The Hilbert curve map is not a homeomorhpism, so it does not preserve topological dimension. The topological dimension and Hausdorff

dimension of the image of the Hilbert map in R' are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a

set in R ) is 1.
[5] Clifford A. Pickover (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (http://

books. google.com/?id=JrslMKTgSZwC&pg=PA3 10&dq=fractal+koch+curve+book#v=onepage&q=fractal koch curve book&f=false).

Sterling Publishing Company, Inc.. p. 310. ISBN 9781402757969. . Retrieved 2011-02-05.
[6] Michael Batty (1985-04-04). "Fractals - Geometry Between Dimensions" (http://books. google. com/?id=sz6GAq_PsmMC&pg=PA31&

dq=mandelbrot+how+long+is+the+coast+of+Britain+Science+book#v=onepage&q=mandelbrot how long is the coast of Britain

Science book&f=false). New Scientist (Holborn Publishing Group) 105 (1450): 31. .

Fractals and Fractional Dimensions 372

[7] John C. Russ (1994). Fractal surfaces, Volume 1 (http://books.google.com/?id=qDQjyuuDRxUC&pg=PAl&lpg=PAl&dq=fractal+

history+book#v=onepage&q=fractal history book&f=false). Springer, p. 1. ISBN 9780306447020. . Retrieved 2011-02-05.
[8] Didier Sornette (2004). Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder : concepts and tools. Springer.

pp. 128-140. ISBN 9783540407546.
[9] Yves Meyer and Sylvie Roques (1993). Progress in wavelet analysis and applications: proceedings of the International Conference

"Wavelets and Applications, " Toulouse, France - June 1992 (http://books. google. com/?id=aHux78oQbbkC&pg=PA25&dq=snowflake+

fractals+book#v=onepage&q=snowflake fractals book&f=false). Atlantica Seguier Frontieres. p. 25. ISBN 9782863321300. . Retrieved

2011-02-05.
[10] Alessandra Carbone, Mikhael Gromov, Przemyslaw Prusinkiewicz (2000). Pattern formation in biology, vision and dynamics (http://books.

google.com/?id=qZHyqUli9y8C&pg=PA78&dq=crystal+fractals+book#v=onepage&q=crystal fractals book&f=false). World Scientific.

p. 78. ISBN 9789810237929. .
[11] Paul S. Addison (1997). Fractals and chaos: an illustrated course (http://books. google. com/?id=12E4ciBQ9qEC&pg=PA45&

dq=lightning+fractals+book#v=onepage&q=lightning fractals book&f=false). CRC Press, pp. 44-46. ISBN 9780750304009. . Retrieved

2011-02-05.
[12] S. V. Buldyrev, A. L. Goldberger, S. Havlin, C. K. Peng and H. E. Stanley (1995). "chapter 3 in A. Bunde and S. Havlin Eds. Fractals in

Science" (http://havlin.biu.ac.il/Shlomo Havlin books_f_in_s.php). Springer. .
[13] "Hunting the Hidden Dimension." Nova. PBS. WPMB-Maryland. 28 October 2008.
[14] Hohlfeld R, Cohen N (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae". Fractals 1 (1):

79-84. doi: 10. 1 142/S02 1 8348X99000098.
[15] "Richard Taylor, Adam P. Micolich and David Jonas. "Fractal Expressionism : Can Science Be Used To Further Our Understanding Of

Art?"" (http://www.phys.unsw.edu.au/PHYSICS_l/FRACTAL_EXPRESSIONISM/fractal_taylor.html). Phys.unsw.edu.au. . Retrieved

2010-10-17.
[16] A Panorama of Fractals and Their Uses (http://classes.yale.edu/Fractals/Panorama/) by Michael Frame and Benoit B. Mandelbrot
[17] "Ron Eglash. "African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999."" (http://

www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm). Rpi.edu. . Retrieved 2010-10-17.
[18] Nelson, Bryn. Sophisticated Mathematics Behind African Village Designs Fractal patterns use repetition on large, small scale (http://

www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2000/02/23/MN36684.DTL), San Francisco Chronicle, Wednesday,

February 23, 2009.
[19] "David Foster Wallace - Bookworm on KCRW" (http://www.kcrw.com/etc/programs/bw/bw96041 ldavid_foster_wallace). Kcrw.com.

.Retrieved 2010-10-17.
[20] "Applications" (http://library.thinkquest.org/26242/full/ap/ap.html). . Retrieved 2007-10-21.

Further reading

• Barnsley, Michael F., and Hawley Rising. Fractals Everywhere. Boston: Academic Press Professional, 1993.
ISBN 0-12-079061-0

• Falconer, Kenneth. Techniques in Fractal Geometry. John Wiley and Sons, 1997. ISBN 0-471-92287-0

• Jiirgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New
York: Springer- Verlag, 1992. ISBN 0-387-97903-4

• Benoit B. Mandelbrot The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. ISBN
0-7167-1186-9

• Peitgen, Heinz-Otto, and Dietmar Saupe, eds. The Science of Fractal Images. New York: Springer- Verlag, 1988.
ISBN 0-387-96608-0

• Clifford A. Pickover, ed. Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of
Advanced Research. Elsevier, 1998. ISBN 0-444-50002-2

• Jesse Jones, Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.

• Hans Lauwerier, Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt,
Princeton University Press, Princeton NJ, 1991. ISBN 0-691 -0855 1-X, cloth. ISBN 0-691-02445-6 paperback.
"This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.

• Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850839-5
and ISBN 978-0-19-850839-7.

• Bernt Wahl, Peter Van Roy, Michael Larsen, and Eric Kampman Exploring Fractals on the Macintosh (http://
www.fractalexplorer.com), Addison Wesley, 1995. ISBN 0-201-62630-6

• Nigel Lesmoir-Gordon. "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals." ISBN
1-904555-05-5 (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the

Fractals and Fractional Dimensions

373

fractal concept and the Mandelbrot set.

• Gouyet, Jean-Francois. Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. ISBN
2-225-85130-1, and New York: Springer- Verlag, 1996. ISBN 978-0-387-94153-0. Out-of-print. Available in PDF
version at. "Physics and Fractal Structures" (http://www.jfgouyet.fr/fractal/fractauk.html) (in (French)).
Jfgouyet.fr. Retrieved 2010-10-17.

• A. Bunde, S. Havlin (1996). Fractals and Disordered Systems (http://havlin.biu.ac.il/Shlomo Havlin
books_fds.php). Springer.

• A. Bunde, S. Havlin (1995). Fractals in Science (http://havlin.biu.ac.il/Shlomo Havlin books_f_in_s.php).
Springer.

• D. Ben-Avraham and S. Havlin (2000). Diffusion and Reactions in Fractals and Disordered Systems (http://
havlin.biu.ac.il/Shlomo Havlin books_d_r.php). Cambridge University Press.

External links

• Fractals (http://www.dmoz.Org/Science/Math/Chaos_And_Fractals//) at the Open Directory Project

Mandelbrot set

The Mandelbrot set is a particular
mathematical set of points, whose
boundary generates a distinctive and
easily recognisable two-dimensional
fractal shape. The set is closely related
to the Julia set (which generates
similarly complex shapes), and is
named after the mathematician Benoit
Mandelbrot, who studied and
popularized it.

More technically, the Mandelbrot set is
the set of values of c in the complex
plane for which the orbit of under
iteration of the complex quadratic

polynomial z , = Z + c remains

bounded. That is, a complex number,

c, is part of the Mandelbrot set if, when

starting with z = and applying the iteration repeatedly, the absolute value of z never exceeds a certain number

(that number depends on c) however large n gets.

For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,..., which tends to infinity. As this sequence is

unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = i (where i is defined as i = -1) gives
the sequence 0, i, (-1 + i), -i, (-1 + i), -i, ..., which is bounded and so 2 belongs to the Mandelbrot set.

Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at
increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The
set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies
to the whole set, and not just to its parts.

Initial image of a Mandelbrot set zoom sequence with a continuously coloured
environment

Mandelbrot set

374

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a
complex structure arising from the application of simple rules, and is one of the best-known examples of
mathematical visualization.

History

The Mandelbrot set has its place in
complex dynamics, a field first
investigated by the French
mathematicians Pierre Fatou and
Gaston Julia at the beginning of the
20th century. The first pictures of this
fractal were drawn in 1978 by Robert
Brooks and Peter Matelski as part of a
study of Kleinian groups. On 1
March 1980, at IBM's Thomas J.
Watson Research Center in upstate
New York, Benoit Mandelbrot first saw

a visualization of the set

[3]

*********

************* ************

*********** ************

********

+ *
+ *

***
*** +

*****
+ ** + ijt
*****

*****

*** +
***

t **
t **

t*
****

****

***

****
**

The first picture of the Mandelbrot set, by Robert Brooks and Peter Matelski in 1978

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980. The
mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H.
Hubbard, who established many of its fundamental properties and named the set in honour of Mandelbrot.

The mathematicians Heinz-Otto Peitgen and Peter Richter became well-known for promoting the set with stunning
photographs, books, and an internationally touring exhibit of the German Goethe-Institut.

The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set
to a wide audience. The cover featured an image created by Peitgen, et al.

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract
mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of
all the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this
article, but such a list would notably include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro

Shishikura, and Jean-Christophe Yoccoz.

Mandelbrot set

375

Formal definition

The Mandelbrot set M is defined by a family of complex quadratic polynomials

P c : C -> C
given by

P c : z i-> z 2 + c,
where c is a complex parameter. For each c , one considers the behavior of the sequence

{0,P c (0),P c (P c (0)),P c (P c (P c (0))),...)

obtained by iterating P c (z) starting at critical point % = 0> which either escapes to infinity or stays within a disk

of some finite radius. The Mandelbrot set is defined as the set of all points c such that the above sequence does not

escape to infinity.

More formally, if P n {z) denotes the

nth iterate of P c (2:)(i.e. P c {z)

composed with itself n times), the

Mandelbrot set is the subset of the

complex plane given by

A mathematician's depiction of the Mandelbrot set M. A point c is coloured black if it
belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts

of c, respectively.

M = {c G C : 3s G R, Mn G N, |f^(0)| < s} .
As explained below, it is in fact possible to simplify this definition by taking g = 2 •

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M
or it does not. A picture of the Mandelbrot set can be made by colouring all the points c which belong to M black,
and all other points white. The more colourful pictures usually seen are generated by colouring points not in the set
according to how quickly or slowly the sequence IP'YO)! diverges to infinity. See the section on computer
drawings below for more details.

The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials P c (z) ■ That is, it
is the subset of the complex plane consisting of those parameters c for which the Julia set of P c is connected.

Mandelbrot set

376

Basic properties

The Mandelbrot set is a compact set, contained in the closed disk of radius 2 around the origin. In fact, a point c
belongs to the Mandelbrot set if and only if

1^(0)1 < 2foralln> 0.
In other words, if the absolute value of i- , "(0)ever becomes larger than 2, the sequence will escape to infinity.
The intersection of J^f with the real axis is precisely the interval [-2,
0.25]. The parameters along this interval can be put in one-to-one
correspondence with those of the real logistic family,

Correspondence between the Mandelbrot set and
the logistic map

z^Az(l-z), AG [1,4].
The correspondence is given by

In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the
Mandelbrot set.

The area of the Mandelbrot set is estimated to be 1.50659177 ± 0.00000008. [13]

Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit
conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk.
Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on
computer pictures generated by programs which are unable to detect the thin filaments connecting different parts of
M . Upon further experiments, he revised his conjecture, deciding that M should be connected.

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and

Hubbard's proof of the connectedness of M > gives rise to external rays of the Mandelbrot set. These rays can be

1141
used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of
parameters c for which the dynamics changes abruptly under small changes of c. It can be constructed as the limit
set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial

lemniscates. The Mandelbrot curves are defined by setting p =z, p =p +z, and then interpreting the set of points
Ip (z)l=2 in the complex plane as a curve in the real Cartesian plane of degree 2 n in x and y.

Mandelbrot set

377

Other properties

The main cardioid and period bulbs

Upon looking at a picture of the Mandelbrot set, one immediately
notices the large cardioid-shaped region in the center. This main
cardioid is the region of parameters cfor which .P c has an attracting
fixed point. It consists of all parameters of the form

2 V 2,
for some /i in the open unit disk.

To the left of the main cardioid, attached to it at the point c = —3/4, a circular-shaped bulb is visible. This bulb
consists of those parameters cfor which .P c has an attracting cycle of period 2. This set of parameters is an actual
circle, namely that of radius 1/4 around -1,
There are infinitely many other bulbs tangent to the main cardioid: for every rational number - with p and q

coprime, there is such a bulb that is tangent at the parameter
' E 'l

2vri'

e i

27ri2

e i

Cp =

Attracting cycle in 2/5-bulb plotted over Julia set
(animation)

■p

This bulb is called the --bulb of the Mandelbrot set. It consists of parameters which have an attracting cycle of

period (J and combinatorial rotation number - . More precisely, the q periodic Fatou components containing the

attracting cycle all touch at a common point (commonly called the a -fixed point). If we label these components
Uq, . . . , U q _iin counterclockwise orientation, then J^maps the component Uj to the component Uj^~ ( mo dn) ■

Mandelbrot set

378

Attracting cycles and Julia sets for parameters in the 1/2, 3/7, 2/5,
1/3, 1/4, and 1/5 bulbs

Cycle periods and antennae

Hyperbolic components

All the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the
maps ,P c have an attracting periodic cycle. Such components are called hyperbolic components.

It is conjectured that these are the only interior regions of M . This problem, known as density of hyperbolicity, may
be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of
the Mandelbrot set are often referred to as "queer" components.

For real quadratic polynomials, this question was answered positively in the 1990s independently by Lyubich and by
Graczyk and Swiatek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic
windows in the Feigenbaum diagram. So this result states that such windows exist near every parameter in the
diagram.)

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the
Mandelbrot set. However, such a component can be reached by a sequence of direct bifurcations from the main
cardioid of a little Mandelbrot copy (see below).

Each of the hyperbolic components has a centre, namely the point c such that the inner Fatou domain for P c (z) has
a super-attracting cycle (the attraction is infinite). This means that the cycle contains the critical point 0, so that is
iterated back to itself after some iterations. We therefore have that P c (0) = Ofor some n. If we call this

polynomial Q n (c), we have that Q n+1 (c) = Q n (c) 2 + land that the degree of Q™(c)is 2 n ~ 1 - We can
therefore construct the centres of the hyperbolic components, by successive solvation of the equations
Q n (c) = 0, n = 1, 2, 3, ...• Note that for each step, we get just as many new centres as we have found so far.

Local connectivity

It is conjectured that the Mandelbrot set is locally connected. This
famous conjecture is known as MLC (for Mandelbrot Locally
Connected). By the work of Adrien Douady and John H. Hubbard, this
conjecture would result in a simple abstract "pinched disk" model of
the Mandelbrot set. In particular, it would imply the important
hyperbolicity conjecture mentioned above.

The work of Jean-Christophe Yoccoz established local connectivity of
the Mandelbrot set at all finitely -renormalizable parameters; that is,
roughly speaking those which are contained only in finitely many
small Mandelbrot copies. Since then, local connectivity has been
proved at many other points of M , but the full conjecture is still
open.

Topological model of Mandelbrot set without

mini Mandlebrot sets and Misiurewicz points (

Cactus model)

Mandelbrot set

379

Thurston model of Mandelbrot set (abstract
Mandelbrot set)

Self-similarity

The Mandelbrot set is self-similar under
magnification in the neighborhoods of the
Misiurewicz points. It is also conjectured to be
self-similar around generalized Feigenbaum
points (e.g. -1.401155 or -0.1528 + 1.0397/), in
the sense of converging to a limit set.

Self similarity in the Mandelbrot set shown by zooming
in on a round feature while panning in the negative-*

direction. The display center pans from (— 1, 0) to
(—1.31, 0) while the view magnifies from 0.5 x 0.5 to
0.12 x 0.12 to approximate the Feigenbaum ratio fi .

Self-similarity around Misiurewicz point -0.1011 + 0.9563i.

Mandelbrot set

380

■ 1 ■"■'''-■ ■"' J"

Quasi-self-similarity in the Mandelbrot set

because of the thin threads connecting them to the main body of the set.

The Mandelbrot set in general is not
strictly self-similar but it is
quasi-self-similar, as small slightly
different versions of itself can be found
at arbitrarily small scales.

The little copies of the Mandelbrot set
are all slightly different, mostly

Further results

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro

1171
Shishikura. It is not known whether the boundary of the Mandelbrot set has positive planar Lebesgue measure.

In the Blum-Shub-Smale model of real computation, the Mandelbrot set is not computable, but its complement is
computably enumerable. However, many simple objects (e.g., the graph of exponentiation) are also not computable
in the BSS model. At present it is unknown whether the Mandelbrot set is computable in models of real computation
based on computable analysis, which correspond more closely to the intuitive notion of "plotting the set by a
computer." Herding has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is
true.

A zoom into the Mandelbrot set illustrating a

Julia "island" and the corresponding Julia set of

the form f (z) = Z ~\~ C > m which c is

the centre of the Mandelbrot set zoom in.

Relationship with Julia sets

As a consequence of the definition of the Mandelbrot set, there is a
close correspondence between the geometry of the Mandelbrot set at a
given point and the structure of the corresponding Julia set.

This principle is exploited in virtually all deep results on the
Mandelbrot set. For example, Shishikura proves that, for a dense set of
parameters in the boundary of the Mandelbrot set, the Julia set has
Hausdorff dimension two, and then transfers this information to the
parameter plane. Similarly, Yoccoz first proves the local connectivity
of Julia sets, before establishing it for the Mandelbrot set at the
corresponding parameters. Adrien Douady phrases this principle as

Plough in the dynamical plane, and harvest in parameter
space.

Mandelbrot set 381

Map of Julia sets for points on the complex plane.

The overall structure, in terms of which Julia sets

are connected, resembles a Mandelbrot set.

Geometry

Recall that, for every rational number - where p and q are relatively prime, there is a hyperbolic component of

period q bifurcating from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this
bifurcation point is called the plq-Mmb. Computer experiments suggest that the diameter of the limb tends to zero
like -j. The best current estimate known is the famous Yoccoz-inequality, which states that the size tends to zero

like -.
i

A period-g limb will have q - 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb
by counting these antennas.

In an attempt to demonstrate that the thickness of the p/q-limb is zero, David Boll carried out a computer experiment
in 1991, where he computed the number of iterations required for the series to converge for z= — j -\- ie{ — §

being the location thereof). As the series doesn't converge for the exact value of z = — ^, the number of iterations

required increases with a small 8. It turns out that multiplying the value of e with the number of iterations required
yields an approximation of jt that becomes better the smaller e. For example, for e = 0.0000001 the number of
iterations is 31415928 and the product is 3.1415928. [18]

Image gallery of a zoom sequence

The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called "zooming
in". The following example of an image sequence zooming to a selected c value gives an impression of the infinite
richness of different geometrical structures, and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10,000,000,000 to 1. Relating to an ordinary
monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometres. Its border would show an
astronomical number of different fractal structures.

Mandelbrot set

382

Start. Mandelbrot set with

continuously coloured

environment.

Gap between the "head" and the

"body" also called the "seahorse

valley".

On the left double-spirals, on the
right "seahorses".

Seahorse" upside down.

The seahorse "body" is composed by 25 "spokes" consisting of 2 groups of 12 "spokes" each and one "spoke"
connecting to the main cardioid. These 2 groups can be attributed by some kind of metamorphosis to the 2 "fingers"
of the "upper hand" of the Mandelbrot set, therefore, the number of "spokes" increases from one "seahorse" to the
next by 2; the "hub" is a so-called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted
small copy of the Mandelbrot set called satellite may be recognized.

The central endpoint of the

"seahorse tail" is also a

Misiurewicz point.

Part of the "tail" — there is only

one path consisting of the thin

structures that lead through the

whole "tail". This zigzag path

passes the "hubs" of the large

objects with 25 "spokes" at the

inner and outer border of the

"tail"; thus the Mandelbrot set is

a simply connected set, which

means there are no islands and no

loop roads around a hole.

Satellite. The two "seahorse tails"

are the beginning of a series of

concentric crowns with the

satellite in the center. Open this

location in an interactive viewer.
[19]

Each of these crowns consists of

similar "seahorse tails"; their

number increases with powers of

2, a typical phenomenon in the

environment of satellites. The

unique path to the spiral center

passes the satellite from the
groove of the cardioid to the top
of the "antenna" on the "head".

"Antenna" of the satellite.

Several satellites of second order

may be recognized.

The "seahorse valley" of the
satellite. All the structures from
the start of the zoom reappear.

' ' M,. .. ■■"■%

Double-spirals and "seahorses" -

unlike the 2nd image from the

start they have appendices

consisting of structures like

"seahorse tails"; this

demonstrates the typical linking

of n+1 different structures in the

environment of satellites of the

order n, here for the simplest case

n=l.

Double-spirals with satellites of

second order - analog to the

"seahorses" the double-spirals

may be interpreted as a

metamorphosis of the "antenna".

Mandelbrot set

383

In the outer part of the
appendices islands of structures
may be recognized; they have a

shape like Julia sets J ; the

largest of them may be found in

the center of the "double-hook"

on the right side.

The islands above seem to consist of infinitely many parts like Cantor sets, as is actually the case for the
corresponding Julia set / . However they are connected by tiny structures so that the whole represents a simply
connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this
magnification. The value of c for the corresponding J is not that of the image center but, relative to the main body of
the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 7th zoom
step.

Generalizations

Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial
family of recursions

Z 1-5- Z d + C.

For integer d, these sets are connectedness loci for the Julia sets built from the same formula. The full cubic
connectedness map has also been studied; here one considers the two-parameter recursion z \— > z 3 + 3kz + c >
whose two critical points are the complex square roots of the parameter k. A point is in the map if either critical point
is stable.

For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation
locus, which is a natural object to study even when the connectedness locus is not useful.

Other non-analytic mappings

Of particular interest is the tricorn fractal, the connectedness locus of the anti-holomorphic family

The tricorn (also sometimes called the Mandelbar set) was encountered by Milnor in his study of parameter slices of
real cubic polynomials. It is not locally connected. This property is inherited by the connectedness locus of real cubic
polynomials.

Another non-analytic generalization is the Burning Ship fractal which is obtained by iterating the mapping

z^(|sft{z)|+i|3(z)|) 2 + c.
The Multibrot set is obtained by varying the value of the exponent d. The article has a video that shows the
development from d = to 7 at which point there are 6 i.e. (d - 1) lobes around the perimeter. A similar development
with negative exponents results in (1 - d) clefts on the inside of a ring.

Mandelbrot set

384

Still image of a movie of increasing

magnification on 0.001643721971153 +

0.822467633298876i

Computer drawings

There are many programs used to generate the Mandelbrot set and
other fractals, some of which are described in fractal-generating
software. These programs use a variety of algorithms to determine the
color of individual pixels and achieve efficient computation.

Escape time algorithm

The simplest algorithm for generating a representation of the
Mandelbrot set is known as the "escape time" algorithm. A repeating
calculation is performed for each x, y point in the plot area and based
on the behaviour of that calculation, a colour is chosen for that pixel.

The x and y locations of each point are used as starting values in a
repeating, or iterating calculation (described in detail below). The

result of each iteration is used as the starting values for the next. The values are checked during each iteration to see
if they have reached a critical 'escape' condition or 'bailout'. If that condition is reached, the calculation is stopped,
the pixel is drawn, and the next x, y point is examined. For some starting values, escape occurs quickly, after only a
small number of iterations. For starting values very close to but not in the set, it may take hundreds or thousands of
iterations to escape. For values within the Mandelbrot set, escape will never occur. The programmer or user must
choose how much iteration, or 'depth,' they wish to examine. The higher the maximum number of iterations, the
more detail and subtlety emerge in the final image, but the longer time it will take to calculate the fractal image.

Escape conditions can be simple or complex. Because no complex number with a real or imaginary part greater than
2 can be part of the set, a common bailout is to escape when either coefficient exceeds 2. A more computationally
complex method, but which detects escapes sooner, is to compute the distance from the origin using the Pythagorean
theorem, and if this distance exceeds two, the point has reached escape. More computationally intensive rendering
variations such as Buddhabrot detect an escape, then use values iterated along the way.

The colour of each point represents how quickly the values reached the escape point. Often black is used to show
values that fail to escape before the iteration limit, and gradually brighter colours are used for points that escape.
This gives a visual representation of how many cycles were required before reaching the escape condition.

For programmers

The definition of the Mandelbrot set, together with its basic properties, suggests a simple algorithm for drawing a
picture of the Mandelbrot set. The region of the complex plane we are considering is subdivided into a certain
number of pixels. To color any such pixel, let cbe the midpoint of that pixel. We now iterate the critical point
under P , checking at each step whether the orbit point has modulus larger than 2.

When this is the case, we know that c does not belong to the Mandelbrot set, and we color our pixel according to the
number of iterations used to find out. Otherwise, we keep iterating up to a fixed number of steps, after which we
decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

In pseudocode, this algorithm would look as follows. The algorithm does not use complex numbers, and manually
simulates complex number operations using two real numbers, for those who do not have a complex data type. If you
have a complex data type in your programming language, using it can simplify your program.

For each pixel on the screen do:

{

xO = scaled x co-ordinate of pixel (must be scaled to lie somewhere in the interval (-2.5 to 1)
yO = scaled y co-ordinate of pixel (must be scaled to lie somewhere in the interval (-1, 1)

Mandelbrot set 385

x =
y =

iteration =
max_iteration = 1000

while ( x*x + y*y <= (2*2) AND iteration < max_iteration )
{

xtemp = x*x - y*y + xO

y = 2*x*y + yO

x = xtemp

iteration = iteration + 1

if ( iteration == max_iteration )
then

color = black
else

color = iteration

plot (x0,y0, color)

where, relating the pseudocode to c , Z and P c :

• z = x + iy

' z = x + i2xy — y

• c = x + iy

and so, as can be seen in the pseudocode in the computation of x and y:

• x = Re(z 2 + c) = x 2 - y 2 + x and V = Im(z 2 + c) = 2xy + y .

To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be
made using one of a variety of functions (linear, exponential, etc.)- One practical way to do it, without slowing down
the calculations, is to use the number of executed iterations as an entry to a look-up color palette table initialized at
startup. If the color table has, for instance, 500 entries, then you can use n mod 500, where n is the number of
iterations, to select the color to use. You can initialize the color palette matrix in various different ways, depending
on what special feature of the escape behavior you want to emphasize graphically.

Continuous (smooth) coloring

Mandelbrot set 386

This image was rendered with the Escape Time Algorithm. Notice the very obvious "bands" of color.

This image was rendered with the Normalized Iteration Count Algorithm. Notice the bands of color have been replaced by a smooth gradient. Also,
the colors take on the same pattern that would be observed if the Escape Time Algorithm was used.

The Escape Time Algorithm is popular for its simplicity. However, it creates bands of color, which, as a type of
aliasing, can detract from an image's aesthetic value. This can be improved using an algorithm known as

[21] [221

"Normalized Iteration Count", which provides a smooth transition of colors between iterations. The algorithm

associates a real number v with each value of z by using the connection of the iteration number with the potential
function. This function is given by

4>(z)=\im(l g\z n \/P n ),

where z is the value after n iterations and P is the power for which z is raised to in the Mandelbrot set equation
(z = z + c, P is generally 2).

If we choose a large bailout radius N (e.g. 10 ), we have that

log\z n \/P n = log(N)/P< z \
for some real number u(z), and this is

v(z) =n- log P (log \z n \j log(iV)),
and as n is the first iteration number such that \z I > N, the number we subtract from n is in the interval [0, 1).

For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H
colours numbered from to H - 1 (H = 500, for instance). We multiply the real number viz) by a fixed real
number determining the density of the colours in the picture, and take the integral part of this number modulo H.

Distance estimates

One can compute the distance from point c (in exterior or interior) to nearest point on the boundary of Mandelbrot

set. [23]

Exterior distance estimation

The proof of the connectedness of the Mandelbrot set in fact gives a formula for the uniformizing map of the
complement of J^f (and the derivative of this map). By the Koebe 1/4 theorem, one can then estimate the distance
between the mid-point of our pixel and the Mandelbrot set up to a factor of 4.

In other words, provided that the maximal number of iterations is sufficiently high, one obtains a picture of the
Mandelbrot set with the following properties:

1 . Every pixel which contains a point of the Mandelbrot set is colored black.

2. Every pixel which is colored black is close to the Mandelbrot set.

Mandelbrot set

387

The distance estimate b of a pixel c (a complex number) from the
Mandelbrot set is given by

Exterior distance estimate may be used to color
whole complement of Mandelbrot set

lim 2-

71— J-OO

i?(c) | -In | i?(c)
I —P n (c) I

where

• P c ( z ) stands for complex quadratic polynomial

• P n (c) stands for n iterations of P c (z) — > Z or 2 2 + c > z> starting with z = C : P°(c) = C .

• — P n (c) i s ^e derivative of P n (c) with respect to c. This derivative can be found by starting with

— P°(c) = l and men — P n+1 (c) = 2 ■ P n (c) ■ — P n (c) + 1- This can easily be verified by using the

chain rule for the derivative.
The idea behind this formula is simple: When the equipotential lines for the potential function ^(z)lie close, the

number |0'(z)| is large, and conversely, therefore the equipotential lines for the function (j)(z)j\(j)'(z)\ should

lie approximately regularly.

From a mathematician's point of view, this formula only works in limit where n goes to infinity, but very reasonable

estimates can be found with just a few additional iterations after the main loop exits.

Once b is found, by the Koebe 1/4-theorem, we know there's no point of the Mandelbrot set with distance from c
smaller than b/4.

The distance estimation can be used for drawing of the boundary of the Mandelbrot set, see the article Julia set.

Mandelbrot set

388

Interior distance estimation

It is also possible to estimate the distance of a limitly periodic (i.e.,
inner) point to the boundary of the Mandelbrot set. The estimate is
given by

Pixels colored according to the estimated interior
distance

- |

&m*>) r

1 d

1 dc

PZ{zo)

dzdz Pc ( Z o) 1

z 2 + c

where

• pis the period,

• c is the point to be estimated,

• i-^fz) is the complex quadratic polynomial P c (z)

• Pc(zo) is the p-fold iteration of P c (z) — > z , starting with P c °(z) = z

• Zqis any of the p points that make the attract or of the iterations of PJz) — > Z starting with P°(z) = C > z

satisfies z n = P* (z n ),
' — — P^(zo). —-PP(zo), —PP(zo) and — Pf(z ) are various derivatives of PP( Z ), evaluated at

Zq.
Analogous to the exterior case, once b is found, we know that all points within the distance of blA from c are inside
the Mandelbrot set.

There are two practical problems with the interior distance estimate: first, we need to find Zoprecisely, and second,
we need to find ^precisely. The problem with Zois that the convergence to Zrjby iterating P c (z) requires,
theoretically, an infinite number of operations. The problem with period is that, sometimes, due to rounding errors, a
period is falsely identified to be an integer multiple of the real period (e.g., a period of 86 is detected, while the real
period is only 43=86/2). In such case, the distance is overestimated, i.e., the reported radius could contain points
outside the Mandelbrot set.

Mandelbrot set

389

Optimizations

One way to improve calculations is to find out beforehand whether the
given point lies within the cardioid or in the period-2 bulb. Before
passing the complex value through the escape time algorithm, first
check if:

3D view : smallest absolute value of the orbit of
the interior points of the Mandelbrot set

+ y 2

x < p - 2p 2 +

(x + ir+y 2 <-

where x represents the real value of the point and y the imaginary value. The first two equations determine if the
point is within the cardioid, the last the period-2 bulb.

The cardioid test can equivalently be performed without the square root:

«(* + (*-i)) < i»'-

[24]

3rd- and higher-order buds do not have equivalent tests, because they are not perfectly circular.

To prevent having to do huge numbers of iterations for other points in the set, one can do "periodicity checking";
which means check if a point reached in iterating a pixel has been reached before. If so, the pixel cannot diverge, and
must be in the set. This is most relevant for fixed-point calculations, where there is a relatively high chance of such
periodicity — a full floating-point (or higher-accuracy) implementation would rarely go into such a period.

Periodicity checking is, of course, a trade-off. The need to remember points costs memory and data management
instructions, whereas it saves computational instructions.

Popular culture

• The Mandelbrot set shape was used by Heart as artwork for their 2004 album, Jupiter's Darling.

• The Jonathan Coulton song, "Mandelbrot Set", is a tribute to both the fractal itself, and to its father Benoit
Mandelbrot. However, the definition given in the song describes the orbit of some arbitrary point on the complex
plane, instead of the orbit of 0.

• The second book of the Mode series, Fractal Mode, describes an entire world that's a perfect 3d model of the set.

• In The Ghost from the Grand Banks by Arthur C. Clarke, the Craig family is obsessed with the Mandelbrot set,

1251
even having a pond in its shape dug. However Clarke's descriptions of the M-set are incorrect.

• Fractals are referenced by Blue Man Group in their stage performances and used in song names on their album
Audio and two of the songs have "Mandelbrot" in the title.

Mandelbrot set 390

• The alternative cover to Outkast's 2000 album Stankonia features the Mandelbrot set.

• The movie Blueberry employs fractal animations in the title sequence and throughout the movie as scene change
intermissions. The movie director Jan Kounen is generally fond of psychedelia.

• The Mandelbrot set is found on the wings of the fictional Quantum Weather Butterfly in Terry Pratchett's
Discworld series.

References

[I] "Mandelbrot Set Explorer: Mathematical Glossary" (http://math.bu.edu/DYSYS/explorer/def.html). . Retrieved 2007-10-07.

[2] Robert Brooks and Peter Matelski, The dynamics of2-generator subgroups of PSL(2,C), in "Riemann Surfaces and Related Topics", ed. Kra

and Maskit, Ann. Math. Stud. 97, 65-71, ISBN 0-691-08264-2
[3] R.P. Taylor & J.C. Sprott (2008). "Biophilic Fractals and the Visual Journey of Organic Screen-savers" (http://sprott.physics.wisc.edu/

pubs/paper311.pdf) (pdf). Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 12, No. 1. Society for Chaos Theory in Psychology &

Life Sciences. . Retrieved 1 January 2009.

[4] Benoit Mandelbrot, Fractal aspects of the iteration of z I — > Az(l — z)for complex A, Z , Annals NY Acad. Sci. 357, 249/259
[5] Adrien Douady and John H. Hubbard, Etude dynamique des polynomes complexes, Prepublications mathemathiques d'Orsay 2/4 (1984 /

1985)
[6] Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer-Verlag. ISBN 0-387-15851-0.
[7] Frontiers of Chaos, Exhibition of the Goethe-Institut by H.O. Peitgen, P. Richter, H. Jurgens, M. Prilfer, D.Saupe. since 1985 shown in over

40 countries.
[8] Gleick, James (1987). Chaos: Making a New Science. London: Cardinal, pp. 229.
[9] Dewdney, A.K. (1985). A computer microscope zooms in for a close look at the most complicated object in mathematics. Scientific

American, pp. 16—24.
[10] Fractals: The Patterns of Chaos. John Briggs. 1992. p. 80.

[II] Lyubich, Mikhail (May— June, 1999). Six Lectures on Real and Complex Dynamics (http://citeseer.ist.psu.edu/cache/papers/cs/28564/
http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/). . Retrieved 2007-04-04.

[12] Lyubich, Mikhail (November 1998). "Regular and stochastic dynamics in the real quadratic family" (http://www.pnas.org/cgi/reprint/

95/24/14025. pdf) (PDF). Proceedings of the National Academy of Sciences of the United States of America 95 (24): 14025-14027.

doi:10.1073/pnas.95.24.14025. PMC 24319. PMID 9826646. . Retrieved 2007-04-04.
[13] Mrob.com (http://www.mrob.com/pub/muency/pixelcounting.html)
[14] The Mandelbrot set, theme and variations. Tan, Lei. Cambridge University Press, 2000. ISBN 9780521774765. Section 2.1, "Yoccoz

para-puzzles", p. 121 (http://books. google. com/books ?id=-a_DsYXquVkC&pg=PA121)
[15] Lei. pdf (http://projecteuclid.Org/euclid.cmp/l 104201823) Tan Lei, "Similarity between the Mandelbrot set and Julia Sets",

Communications in Mathematical Physics 134 (1990), pp. 587-617.
[16] J. Milnor, "Self-Similarity and Hairiness in the Mandelbrot Set", in Computers in Geometry and Topology, M. Tangora (editor), Dekker,

New York, pp. 211-257.
[17] Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. 147 (1998) p. 225-267.

(First appeared in 1991 as a Stony Brook IMS Preprint (http://www.math.sunysb.edu/preprints.html), available as

arXiv:math.DS/9201282 (http://www.arxiv.org/abs/math.DS/9201282).)
[18] Gary William Flake, The Computational Beauty of Nature, 1998. p. 125. ISBN 978-0-262-56127-3
[19] http://guciek.net/web_mandelbrot/en#-0.743643;0.131825;0.00003;5000

[20] Rudy Rucker's discussion of the CCM: CS.sjsu.edu (http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm)
[21] Garcia, Francisco; Angel Fernandez, Javier Barrallo, Luis Martin (PDF). Coloring Dynamical Systems in the Complex Plane (http://math.

unipa.it/~grim/Jbarrallo.PDF). . Retrieved 2008-01-21.
[22] Linas Vepstas. "Renormalizing the Mandelbrot Escape" (http://linas.org/art-gallery/escape/escape.html). .

[23] Albert Lobo Cusido. "Interior and exterior distance bounds for the Mandelbrot" (http://usuarios.lycos.es/llopsite/BuddhaBrotA2.htm). .
[24] "Mandelbrot Bud Maths" (http://linas.org/art-gallery/bud/bud.html). .
[25] Kasman, Alex. "MathFiction: The Ghost from the Grand Banks (Arthur C. Clarke)" (http://kasmana.people.cofc.edu/MATHFICT/

mfview.php?callnumber=mfl2). . Retrieved 2010-10-18.

Mandelbrot set

391

Further reading

• John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160,
(Princeton University Press, 2006), ISBN 0-691-12488-4

(First appeared in 1990 as a Stony Brook IMS Preprint (http://www.math.sunysb.edu/preprints.html),
available as arXiV:math.DS/9201272 (http://www.arxiv.org/abs/math.DS/9201272) )

• Nigel Lesmoir-Gordon, The Colours of Infinity: The Beauty, The Power and the Sense of Fractals, ISBN
1-904555-05-5

(includes a DVD featuring Arthur C. Clarke and David Gilmour)

• Heinz-Otto Peitgen, Hartmut Jiirgens, Dietmar Saupe, Chaos and Fractals - New Frontiers of Science (Springer,
New-York, 1992, 2004), ISBN 0-387-20229-3

• Eric Baird, Alt. Fractals: A visual guide to fractal geometry and design (Chocolate Tree Books, Brighton, 201 1),
ISBN 0955706831

External links

• Chaos and Fractals (http://www.dmoz.org/Science/Math/Chaos_and_Fractals/) at the Open Directory Project

• The Mandelbrot Set and Julia Sets by Michael Frame, Benoit Mandelbrot, and Nial Neger (http://classes.yale.
edu/Fractals/MandelSet/welcome.html)

• For Fractal Design and Consultancy (http://www.fractal.org)

• Mandelbulb/Juliabulb/Juliusbulb (http://www.fractal.org/Mandelbulb.pdf)

• Video: Mandelbrot fractal zoom to 6.066 e228 (http://vimeo.com/12185093)

• H"idd«)«r (mandalabeth) (http://www-personal.umich.edu/~bethchen/mandalabeth/) 3D analog of the
mandelbrot set, with various symmetry groups

Julia set

In the context of complex dynamics, a topic of
mathematics, the Julia set and the Fatou set are two

complementary sets defined from a function.
Informally, the Fatou set of the function consists of
values with the property that all nearby values behave
similarly under repeated iteration of the function, and
the Julia set consists of values such that an arbitrarily
small perturbation can cause drastic changes in the
sequence of iterated function values. Thus the
behavior of the function on the Fatou set is 'regular',
while on the Julia set its behavior is 'chaotic'.

The Julia set of a function f is commonly denoted 7(f),
and the Fatou set is denoted F(f). These sets are
named after the French mathematicians Gaston
Julia and Pierre Fatou, whose work began the study of complex dynamics during the early 20th century.

Julia set 392

Formal definition

Let f(z)be a complex rational map from the plane into itself, that is, f(z) = p(z)/q(z), where p(z) and
g(z)are complex polynomials. Then there are a finite number of open sets F iy i = 1 ; . . . , r , that are left

invariant by f(z) and are such that:

1 . the union of the Fi 's is dense in the plane and

2. / (.z)behaves in a regular and equal way on each of the sets Fi ■

The last statement means that the termini of the sequences of iterations generated by the points of Fi are either
precisely the same set, which is then a finite cycle, or they are finite cycles of finite or annular shaped sets that are
lying concentrically. In the first case the cycle is attracting, in the second it is neutral.

These sets i^are the Fatou domains of f(z), and their union is the Fatou set F(f )of f(z)- Each of the

Fatou domains contains at least one critical point of f(z), that is, a (finite) point z satisfying f'(z) = 0, or z =

oo, if the degree of the numerator p(z)is at least two larger than the degree of the denominator q(z), or if

f(z) = 1/ g(z) -\- cfor some c and a rational function q(z) satisfying this condition.
The complement of i*Y/)is the Julia set J(f )of f(z)- =/(/)i s a nowhere dense set (it is without interior

points) and an uncountable set (of the same cardinality as the real numbers). Like F(f), =/(/)i s left invariant by

f(z), and on this set the iteration is repelling, meaning that \f(z) — f(w)\ > \z — w\ for all w in a

neighbourhood of z (within J(/))- This means that /(z)behaves chaotically on the Julia set. Although there are

points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and

they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave

chaotically, a phenomenon called deterministic chaos.

There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational

maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.

Each component of the Fatou set of a rational map can be classified into one of four different classes.

Equivalent descriptions of the Julia set

• J(/)i s the smallest closed set containing at least three points which is completely invariant under /.

• J(/) is the closure of the set of repelling periodic points.

• For all but at most two points z £ X > the Julia set is the set of limit points of the full backwards orbit

U J \ z ) ■ (This suggests a simple algorithm for plotting Julia sets, see below.)

• If /is an entire function - in particular, when /is a polynomial, then jY/)is the boundary of the set of points

which converge to infinity under iteration.

• If /is a polynomial, then J(/)is the boundary of the filled Julia set; that is, those points whose orbits under

iterations of /remain bounded.

Julia set

393

Properties of the Julia set and Fatou set

The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f ,

and

f-\J(f)) = fW)) = J(f)

r x (F(f)) = f(Hf)) = F(f)

[6]

Examples

For f(z) = z 2 the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation
that is chaotic on the non-rational points). There are two Fatou domains: the interior and the exterior of the circle,
with iteration towards and °°, respectively.

For f(z) = z 2 — 2 the Julia set is the line segment between -2 and 2, and the iteration corresponds to
x — > 4(x — 0.5) 2 i n the unit interval. This can be used as a method for generating pseudorandom numbers. There

is one Fatou domain: the points not on the line segment iterate towards °°.

These two functions are of the form z 2 _|_ c , where c is a complex number. For such an iteration the Julia set is not

in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures

below.

For some functions / (z)we can say beforehand that the Julia set is a

fractal and not a simple curve. This is because of the following main
theorem on the iterations of a rational function:

Each of the Fatou domains has the same boundary, which
consequently is the Julia set

Julia set (in white) for the rational function

associated to Newton's method forf:z— >z -1.

Coloring of Fatou set according to attractor (the

roots of I)

This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if
there are more than two Fatou domains, each point of the Julia set must have points of more than two different open
sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for
instance, when f(z)i & the Newton iteration for solving the equation z n = \{n > 2) :
f(z) = z — f(z)/f'(z) = (1 + (n — l)z"')/(riz"'~ 1 ) ■ The image on the right shows the case n = 3.

Julia set

394

Quadratic polynomials

A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational
maps. The quadratic polynomials can be expressed as

f c (z) = z 2 + c
where c is a complex parameter.

Filled Julia set for f ,

c=l-<p where cp is the
golden ratio

Julia set for f , c=(cp-2)+(cp-l)i
=-0.4+0.6i

Julia set for f , c=0.285+0i

Julia set for f , c=0.45+0.1428i

**&%

1 *&* *m*L* 1

*#+

Julia set for f ,
c
c=-0.70176-0.3842i

Julia set for f c=-0.835-0.2321i

Julia set for f , c=-0.8+0.156i
c

The parameter plane of quadratic polynomials - that is, the plane of
possible c -values - gives rise to the famous Mandelbrot set. Indeed,
the Mandelbrot set is defined as the set of all csuch that l 7(/ c )is
connected. For parameters outside the Mandelbrot set, the Julia set is a
Cantor set: in this case it is sometimes referred to as Fatou dust.

In many cases, the Julia set of c looks like the Mandelbrot set in
sufficiently small neighborhoods of c . This is true, in particular, for
so-called 'Misiurewicz' parameters, i.e. parameters cfor which the
critical point is pre-periodic. For instance:

• At c = i , the shorter, front toe of the forefoot, the Julia set looks
like a branched lightning bolt.

• At c — —2, the tip of the long spiky tail, the Julia set is a straight line segment

A Julia set plot showing julia sets for different
values of c, the plot resembles the Mandelbrot set

In other words the Julia sets JffAaie locally similar around Misiurewicz points

[7]

Julia set 395

Generalizations

The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their
domain; most notably transcendental meromorphic functions and Epstein's 'finite-type maps'.

Julia sets are also commonly defined in the study of dynamics in several complex variables.

The potential function and the real iteration number

The Julia set for f(z) = z 2 is the umt circle, and on the outer Fatou domain, the potential function 0(z)is
defined by 0(z) = log \z\ ■ The equipotential lines for this function are concentric circles. As I f(z)\ = |z| 2we
have 4>{z) = lim log | z k \/2 , where Zfcis the sequence of iteration generated by z. For the more general

k— >oc

iteration f(z) = z 2 + c> it has been proved that if the Julia set is connected (that is, if c belongs to the (usual)

Mandelbrot set), then there exist a biholomorphic map ip between the outer Fatou domain and the outer of the unit

circle such that |^(/(z))| = l^fz)! 2 - This means that the potential function on the outer Fatou domain

\T \ ' ' v 1 1 1 ink :e is given by:
<f)(z) = lim log|z fe |/2 .

k— >oo

This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function
on the Fatou domain containing °° by this formula. For a general rational function f(z)such that » is a critical
point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the
denominator, we define the potential function on the Fatou domain containing °° by:

(j)(z) = lim log \z k \/d k ,

k— >oc

where d = m - n is the degree of the rational function.

If N is a very large number (e.g. 10 ), and if k is the first iteration number such that \zA > N , we have that

log \zu\/d k = log(JV)/d"^> f° r some rea l number v(z), which should be regarded as the real iteration

number, and we have that:

i/(z) = fe-log(log|z fc |/log(JV))/log(cO )
where the last number is in the interval [0, 1).

For iteration towards a finite attracting cycle of order r, we have that if z* is a point of the cycle, then
/(/(.../(z*))) = z*(the r-fold composition), and the number a — l/\(d(f(f(- ■ ■ f (z)))) f dz) Z=Z J\
(> 1) is the attraction of the cycle. If w is a point very near z* and w' is w iterated r times, we have that

a = lim \w — z*\f\w' — z*\. Therefore the number \ z . — z *\a k ^ almost independent of k. We

k— >oo I I

define the potential function on the Fatou domain by:
(j>(z) — lim 1/dzfc,. — z* \a k ).

k— >oc

If e is a very small number and k is the first iteration number such that I z^ — Z * I < e , we have that
(h(z) = l/(ea! l/ '- z ))f or some real number viz), which should be regarded as the real iteration number, and we
have that:

i/(z) =k- log(e/|z fc -z*\)l log(a).
If the attraction is °°, meaning that the cycle is super-attracting, meaning again that one of the points of the cycle is a
critical point, we must replace a. by Oi = lim log \w — z * |/ log \w — Z * \ (where w' is w iterated r times)

k— >oc

and the formula for 0(z)by:

<f)(z) = lim log(l/|z fcr - z* \)/a k .

k— >oo

And now the real iteration number is given by:

v[z) = k- log(log \z k - z * |/log(e))/log(at).

Julia set

396

For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H

colours numbered from to H-l (H = 500, for instance). We multiply the real number i/(z)by a fixed real number

determining the density of the colours in the picture, and take the integral part of this number modulo H.

The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not

neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is

a revolving movement, we can, for instance, colour by the minimum distance from the cycle left fixed by the

iteration.

Field lines

In each Fatou domain (that is not neutral) there are two systems of
lines orthogonal to each other: the equipotential lines (for the potential
function or the real iteration number) and the field lines.

If we colour the Fatou domain according to the iteration number (and
not the real iteration number), the bands of iteration show the course of
the equipotential lines. If the iteration is towards °° (as is the case with
the outer Fatou domain for the usual iteration 2 2 -4- c)' we can easily
show the course of the field lines, namely by altering the colour
according as the last point in the sequence of iteration is above or
below the x-axis (first picture), but in this case (more precisely: when
the Fatou domain is super-attracting) we cannot draw the field lines
coherently - at least not by the method we describe here. In this case a
field line is also called an external ray.

The equipotential lines for iteration towards
infinity

Field lines for an iteration of the form

(l-z 3 /6)/(z-z 2 /2) 2 + c

Let z be a point in the attracting Fatou domain. If we iterate z a large number of times, the terminus of the sequence
of iteration is a finite cycle C, and the Fatou domain is (by definition) the set of points whose sequence of iteration
converges towards C. The field lines issue from the points of C and from the (infinite number of) points that iterate
into a point of C. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let r
be the order of the cycle C (its number of points) and let z* be a point in C. We have /(/(. . . f(z*))) = z* (the
r-fold composition), and we define the complex number a by

a = (d(f(f(...f(z))))/dz) z=z *.
If the points of C are 2j, % = 1 t{z\ = Z*), Ct is the product of the r numbers /'(zj) ■ The real number 1/

let I is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means
that 1 < 1/lal < °°. The point z* is a fixed point for /(/(. . . f(z))), and near this point the map /(/(. . . f(z)))

has (in connection with field lines) character of a rotation with the argument j3 of a (that is, a = \cx\e^ 1 )■

In order to colour the Fatou domain, we have chosen a small number e and set the sequences of iteration

Zk(k = 0, 1, 2, . . . , Zq = z)to stop when \z k — z * I < e , and we colour the point z according to the number

Julia set

397

k (or the real iteration number, if we prefer a smooth colouring). If we choose a direction from z* given by an angle Q ,

the field line issuing from z* in this direction consists of the points z such that the argument ip of the number z^ — Z* satisfies

the condition that

tp — kp = 6 mod 7r.
For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is
increased by 1 and the number if) is increased by j3 , therefore the number ijj — kf3 mod 7T is constant along
the field line.

A colouring of the field lines of the Fatou domain means that we
colour the spaces between pairs of field lines: we choose a number of
regularly situated directions issuing from z*, and in each of these
directions we choose two directions around this direction. As it can
happen that the two field lines of a pair do not end in the same point of
the Julia set, our coloured field lines can ramify (endlessly) in their
way towards the Julia set. We can colour on the basis of the distance to
the centre line of the field line, and we can mix this colouring with the
usual colouring. Such pictures can be very decorative (second picture).

-...

**«.,

®«ftft

Pictures in the field lines for an iteration of the
form — *

Z* +C

A coloured field line (the domain between two field lines) is divided

up by the iteration bands, and such a part can be put into a one-to-one

correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding

field lines, the other is (calculated from) the distance from the inner of the bounding iteration bands (this number is

the non-integral part of the real iteration number). Therefore we can put pictures into the field lines (third picture).

Distance estimation

As a Julia set is infinitely thin we cannot draw it effectively by
backwards iteration from the pixels. It will appear fragmented because
of the impracticality of examining infinitely many startpoints. Since
the iteration count changes vigorously near the Julia set, a partial
solution is to imply the outline of the set from the nearest color
contours, but the set will tend to look muddy.

Julia set drawn by distance estimation, the
iteration is of the form

l-z 2 + z 5 /(2 + Az) + c

Julia set

398

Three-dimensional rendering of Julia set using
distance estimation.

A better way to draw the Julia set in black and white is to estimate the distance of pixels from the set and to color
every pixel whose center is close to the set. The formula for the distance estimation is derived from the formula for
the potential function (j)(z)- When the equipotential lines for <^(,z)lie close, the number \(h (z\\ i s l ar g e > an d
conversely, therefore the equipotential lines for the function S(z) = d)(z)/\(f) (z)\ should lie approximately
regularly. It has been proven that the value found by this formula (up to a constant factor) converges towards the true
distance for z converging towards the Julia set.

We assume that jYzlis rational, that is, f(z) = p(z)/q(z) where p(z)and q(z)are complex polynomials of
degrees m and «, respectively, and we have to find the derivative of the above expressions for (j)(z) ■ And as it is
only Zfc that varies, we must calculate the derivative z' k of Zfc with respect to z. But as z^ = f(f(- * • /(•2 ; )))(the
k-fo\d composition), z' is the product of the numbers f'tz^), and this sequence can be calculated recursively by

zLii = f'(zk)z'u, starting with zL = 1 (before the calculation of the next iteration Zt+i = ffzi.)
For" iteration towanis °° (more precisely when m>n + 2, so that °o is a super-attracting fixed pomr), Vi

e have

|0' (z)\ = lim Kl/N^,
(d= m - n) and consequently:

8(z) - 0(z)/|0'O)| - lim log|z fc ||z fc |/|4|.

fc— >oo

For iteration towards a finite attracting cycle (that is not super-attracting) containing the point z* and having order r,
we have

lira \z\

k— >oc

Zkr

z* | a )

and consequently:

5{z) = (j)(z)/\(j)'(z)\ = lim \z kr - z*

k—>ao

For a super-attracting cycle, the formula is:

6{z)

lim log \z kT

k-^>aa

Z * Zk T

Z *

'fcr-l

'fcrl

We calculate this number when the iteration stops. Note that the distance estimation is independent of the attraction
of the cycle. This means that it has meaning for transcendental functions of "degree infinity" (e.g. sin(z) and tan(z)).

Besides drawing of the boundary, the distance function can be introduced as a 3rd dimension to create a solid fractal
landscape.

Julia set

399

Plotting the Julia set

Using backwards (inverse) iteration (IIM)

As mentioned above, the Julia set can be found as the set of limit
points of the set of pre-images of (essentially) any given point. So we
can try to plot the Julia set of a given function as follows. Start with
any point z we know to be in the Julia set, such as a repelling periodic
point, and compute all pre-images of z under some high iterate f n of

/•

Unfortunately, as the number of iterated pre-images grows
exponentially, this is not feasible computationally. However, we can
adjust this method, in a similar way as the "random game" method for
iterated function systems. That is, in each step, we choose at random
one of the inverse images of f .

For example, for the quadratic polynomial f c , the backwards iteration
is described by

A Julia set plot, generated using MUM

At each step, one of the two square roots is selected at random.

Note that certain parts of the Julia set are quite difficult to access with the reverse Julia algorithm. For this reason,
one must modify IIM/J ( it is called MIIM/J) or use other methods to produce better images.

Julia set 400

Using DEM/J

Julia set : image with C source code using DEM/J

Notes

[1] Note that for other areas of mathematics the notation J( f 1 can also represent the Jacobian matrix of a real valued mapping J between

smooth manifolds.
[2] Gaston Julia (1918) "Memoire sur l'iteration des fonctions rationnelles," Journal de Mathematiques Pures et Appliquees, vol. 8, pages

47-245.
[3] Pierre Fatou (1917) "Sur les substitutions rationnelles," Comptes Rendus de I'Academie des Sciences de Paris, vol. 164, pages 806-808 and

vol. 165, pages 992-995.
[4] Beardon, Iteration of Rational Functions, Theorem 5.6.2
[5] Beardon, Theorem 7.1.1

[6] Beardon, Iteration of Rational Functions, Theorem 3.2.4
[7] Lei.pdf (http://projecteuclid.org/euclid.cmp/1104201823) Tan Lei, "Similarity between the Mandelbrot set and Julia Sets",

Communications in Mathematical Physics 134 (1990), pp. 587—617.
[8] Add en Douady and John H. Hubbard, Etude dynamique des polynomes complexes, Prepublications mathemathiques d'Orsay 2/4 (1984 /

1985)
[9] Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer- Verlag. ISBN 0-387-15851-0.
[10] Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer- Verlag. ISBN 0-387-15851-0.

References

• Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993

• Adrien Douady and John H. Hubbard, "Etude dynamique des polynomes complexes", Prepublications
mathemathiques d'Orsay 2/4 (1984 / 1985)

• John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160,
Princeton University Press 2006 (First appeared in 1990 as a Stony Brook IMS Preprint (http://www.math.
sunysb.edu/preprints.html), available as arXiV:math.DS/9201272 (http://www.arxiv.org/abs/math.DS/
9201272).)

• Alexander Bogomolny, " Mandelbrot Set and Indexing of Julia Sets (http://www.cut-the-knot.org/Curriculum/
Algebra/Julialndexing.shtml)" at cut-the-knot.

• Evgeny Demidov, " The Mandelbrot and Julia sets Anatomy (http://ibiblio.org/e-notes/MSet/Contents.htm)"
(2003)

• Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2

Julia set

401

External links

• Weisstein, Eric W., " Julia Set (http://mathworld.wolfram.com/JuliaSet.html)" from MathWorld.

• Julia Set Fractal (2D) (http://local.wasp.uwa.edu.au/~pbourke/fractals/juliaset/), Paul Burke

• The Julia Set in Four Dimensions (http://www.relativitybook.com/CoolStuff/julia_set_4d.html)

• Julia Sets (http://mathmo.blogspot.com/2007/04/essay-backtrack-julia-sets.html), Jamie Sawyer

• Julia Jewels: An Exploration of Julia Sets (http://mcgoodwin.net/julia/juliajewels.html), Michael McGoodwin

• Crop circle Julia Set (http://www.lucypringle.co.uk/photos/1996/ukl996ck.shtml), Lucy Pringle

• Interactive Julia Set Applet (http://www.cslearn.netne.net/hightechdreams/weaver.php?topic=fractals), Josh
Greig

• Julia and Mandelbrot Set Explorer (http://alephO.clarku.edu/~djoyce/julia/explorer.html), David E. Joyce

• A simple program to generate Julia sets (Windows, 370 kb) (http://www.lizardie.com/links/download/
fractal-generator)

• A collection of applets (http://ifs-tools.sourceforge.net/) one of which can render Julia sets via Iterated
Function Systems.

• Julia meets HTML5 (http://juliamap.googlelabs.com/) Google Labs' HTML5 Fractal generator on your
browser

• Julia (http://cran.r-project.org/web/packages/Julia/index.html) GNU R Package to generate Julia or
Mandelbrot set at a given region and resolution.

Complexity

In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement.
The study of these complex linkages is the main goal of network theory and network science. In science there are
at this time a number of approaches to characterizing complexity, many of which are reflected in this article. In a
business context, complexity management is the methodology to minimize value-destroying complexity and
efficiently control value-adding complexity in a cross-functional approach.

Overview

Definitions are often tied to the
concept of a "system" — a set of parts
or elements that have relationships
among them differentiated from
relationships with other elements
outside the relational regime. Many
definitions tend to postulate or assume
that complexity expresses a condition
of numerous elements in a system and
numerous forms of relationships
among the elements. At the same time,
what is complex and what is simple is
relative and changes with time.

A map of many of the leading scholars and areas of research in complexity science

Some definitions key on the question of the probability of encountering a given condition of a system once
characteristics of the system are specified. Warren Weaver has posited that the complexity of a particular system is

Complexity 402

the degree of difficulty in predicting the properties of the system, if the properties of the system's parts are given. In
Weaver's view, complexity comes in two forms: disorganized complexity, and organized complexity. Weaver's
paper has influenced contemporary thinking about complexity.

The approaches that embody concepts of systems, multiple elements, multiple relational regimes, and state spaces
might be summarized as implying that complexity arises from the number of distinguishable relational regimes (and
their associated state spaces) in a defined system.

Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or
mathematical expression, as is later set out herein.

Disorganized complexity vs. organized complexity

One of the problems in addressing complexity issues has been distinguishing conceptually between the large number
of variances in relationships extant in random collections, and the sometimes large, but smaller, number of
relationships between elements in systems where constraints (related to correlation of otherwise independent
elements) simultaneously reduce the variations from element independence and create distinguishable regimes of
more-uniform, or correlated, relationships, or interactions.

Weaver perceived and addressed this problem, in at least a preliminary way, in drawing a distinction between
"disorganized complexity" and "organized complexity".

In Weaver's view, disorganized complexity results from the particular system having a very large number of parts,
say millions of parts, or many more. Though the interactions of the parts in a "disorganized complexity" situation can
be seen as largely random, the properties of the system as a whole can be understood by using probability and
statistical methods.

A prime example of disorganized complexity is a gas in a container, with the gas molecules as the parts. Some would
suggest that a system of disorganized complexity may be compared, for example, with the (relative) simplicity of the
planetary orbits — the latter can be known by applying Newton's laws of motion, though this example involved highly
correlated events.

Organized complexity, in Weaver's view, resides in nothing else than the non-random, or correlated, interaction
between the parts. These correlated relationships create a differentiated structure that can, as a system, interact with
other systems. The coordinated system manifests properties not carried by, or dictated by, individual parts. The
organized aspect of this form of complexity vis a vis other systems than the subject system can be said to "emerge,"
without any "guiding hand".

The number of parts does not have to be very large for a particular system to have emergent properties. A system of
organized complexity may be understood in its properties (behavior among the properties) through modeling and
simulation, particularly modeling and simulation with computers. An example of organized complexity is a city
neighborhood as a living mechanism, with the neighborhood people among the system's parts.

Sources and factors of complexity

The source of disorganized complexity is the large number of parts in the system of interest, and the lack of
correlation between elements in the system.

There is no consensus at present on general rules regarding the sources of organized complexity, though the lack of
randomness implies correlations between elements. See e.g. Robert Ulanowicz's treatment of ecosystems.
Consistent with prior statements here, the number of parts (and types of parts) in the system and the number of
relations between the parts would have to be non-trivial — however, there is no general rule to separate "trivial" from
"non-trivial".

Complexity 403

Complexity of an object or system is a relative property. For instance, for many functions (problems), such a
computational complexity as time of computation is smaller when multitape Turing machines are used than when
Turing machines with one tape are used. Random Access Machines allow one to even more decrease time
complexity (Greenlaw and Hoover 1998: 226), while inductive Turing machines can decrease even the complexity
class of a function, language or set (Burgin 2005). This shows that tools of activity can be an important factor of
complexity.

Specific meanings of complexity

In several scientific fields, "complexity" has a specific meaning :

• In computational complexity theory, the amounts of resources required for the execution of algorithms is studied.
The most popular types of computational complexity are the time complexity of a problem equal to the number of
steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in
bits), using the most efficient algorithm, and the space complexity of a problem equal to the volume of the
memory used by the algorithm (e.g., cells of the tape) that it takes to solve an instance of the problem as a
function of the size of the input (usually measured in bits), using the most efficient algorithm. This allows to
classify computational problems by complexity class (such as P, NP ... ). An axiomatic approach to computational
complexity was developed by Manuel Blum. It allows one to deduce many properties of concrete computational
complexity measures, such as time complexity or space complexity, from properties of axiomatically defined
measures.

• In algorithmic information theory, the Kolmogorov complexity (also called descriptive complexity, algorithmic
complexity or algorithmic entropy) of a string is the length of the shortest binary program that outputs that string.
Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, monotone
complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity. An axiomatic
approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark Burgin in the
paper presented for publication by Andrey Kolmogorov (Burgin 1982). The axiomatic approach encompasses
other approaches to Kolmogorov complexity. It is possible to treat different kinds of Kolmogorov complexity as
particular cases of axiomatically defined generalized Kolmogorov complexity. Instead, of proving similar
theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such
results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of the
axiomatic approach in mathematics. The axiomatic approach to Kolmogorov complexity was further developed in
the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003).

• In information processing, complexity is a measure of the total number of properties transmitted by an object and
detected by an observer. Such a collection of properties is often referred to as a state.

• In business, complexity describes the variances and their consequences in various fields such as product portfolio,
technologies, markets and market segments, locations, manufacturing network, customer portfolio, IT systems,
organization, processes etc.

• In physical systems, complexity is a measure of the probability of the state vector of the system. This should not
be confused with entropy; it is a distinct mathematical measure, one in which two distinct states are never
conflated and considered equal, as is done for the notion of entropy statistical mechanics.

• In mathematics, Krohn-Rhodes complexity is an important topic in the study of finite semigroups and automata.

• In software engineering, programming complexity is a measure of the interactions of the various elements of the
software. This differs from the computational complexity described above in that it is a measure of the design of
the software.

There are different specific forms of complexity:

Complexity 404

• In the sense of how complicated a problem is from the perspective of the person trying to solve it, limits of
complexity are measured using a term from cognitive psychology, namely the hrair limit.

• Complex adaptive system denotes systems that have some or all of the following attributes

• The number of parts (and types of parts) in the system and the number of relations between the parts is
non-trivial — however, there is no general rule to separate "trivial" from "non-trivial";

• The system has memory or includes feedback;

• The system can adapt itself according to its history or feedback;

• The relations between the system and its environment are non-trivial or non-linear;

• The system can be influenced by, or can adapt itself to, its environment; and

• The system is highly sensitive to initial conditions.

Study of complexity

Complexity has always been a part of our environment, and therefore many scientific fields have dealt with complex
systems and phenomena. From one perspective, that which is somehow complex — displaying variation without
being random — is most worthy of interest given the rewards found in the depths of exploration.

The use of the term complex is often confused with the term complicated. In today's systems, this is the difference
between myriad connecting "stovepipes" and effective "integrated" solutions. This means that complex is the
opposite of independent, while complicated is the opposite of simple.

While this has led some fields to come up with specific definitions of complexity, there is a more recent movement
to regroup observations from different fields to study complexity in itself, whether it appears in anthills, human
brains, or stock markets. One such interdisciplinary group of fields is relational order theories.

Complexity topics
Complex behaviour

The behavior of a complex system is often said to be due to emergence and self-organization. Chaos theory has
investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour.

Complex mechanisms

Recent developments around artificial life, evolutionary computation and genetic algorithms have led to an
increasing emphasis on complexity and complex adaptive systems.

Complex simulations

In social science, the study on the emergence of macro-properties from the micro-properties, also known as
macro-micro view in sociology. The topic is commonly recognized as social complexity that is often related to the
use of computer simulation in social science, i.e.: computational sociology.

Complex systems

Systems theory has long been concerned with the study of complex systems (In recent times, complexity theory and
complex systems have also been used as names of the field). These systems can be biological, economic,
technological, etc. Recently, complexity is a natural domain of interest of the real world socio-cognitive systems and
emerging systemics research. Complex systems tend to be high-dimensional, non-linear and hard to model. In
specific circumstances they may exhibit low dimensional behaviour.

Complexity 405

Complexity in data

In information theory, algorithmic information theory is concerned with the complexity of strings of data.

Complex strings are harder to compress. While intuition tells us that this may depend on the codec used to compress
a string (a codec could be theoretically created in any arbitrary language, including one in which the very small
command "X" could cause the computer to output a very complicated string like "18995316"), any two
Turing-complete languages can be implemented in each other, meaning that the length of two encodings in different
languages will vary by at most the length of the "translation" language — which will end up being negligible for
sufficiently large data strings.

These algorithmic measures of complexity tend to assign high values to random noise. However, those studying
complex systems would not consider randomness as complexity.

Information entropy is also sometimes used in information theory as indicative of complexity.

Applications of complexity

Computational complexity theory is the study of the complexity of problems — that is, the difficulty of solving them.
Problems can be classified by complexity class according to the time it takes for an algorithm — usually a computer
program — to solve them as a function of the problem size. Some problems are difficult to solve, while others are
easy. For example, some difficult problems need algorithms that take an exponential amount of time in terms of the
size of the problem to solve. Take the travelling salesman problem, for example. It can be solved in time 0(n 2 2 71 )
(where n is the size of the network to visit — let's say the number of cities the travelling salesman must visit exactly
once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially.
Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple.
These problems might require large amounts of time or an inordinate amount of space. Computational complexity
may be approached from many different aspects. Computational complexity can be investigated on the basis of time,
memory or other resources used to solve the problem. Time and space are two of the most important and popular
considerations when problems of complexity are analyzed.

There exist a certain class of problems that although they are solvable in principle they require so much time or
space that it is not practical to attempt to solve them. These problems are called intractable.

There is another form of complexity called hierarchical complexity. It is orthogonal to the forms of complexity
discussed so far, which are called horizontal complexity

Bejan and Lorente showed that complexity is modest (not maximum, not increasing), and is a feature of the natural

ro]

phenomenon of design generation in nature, which is predicted by the Constructal law.

Bejan and Lorente also showed that all the optimality (max,min) statements have limited ad-hoc applicability, and
are unified under the Constructal law of design and evolution in nature.

Complexity 406

References

[1] J. M. Zayed, N. Nouvel, U. Rauwald, O. A. Scherman, Chemical Complexity — supramolecular self-assembly of synthetic and biological

building blocks in water, Chemical Society Reviews, 2010, 39, 2806—2816 http://pubs.rsc.org/en/Content/ArticleLanding/2010/CS/

b922348g
[2] Weaver, Warren (1948). "Science and Complexity" (http://www.ceptualinstitute.com/genre/weaver/weaver-1947b.htm). American

Scientist 36 (4): 536. PMID 18882675. . Retrieved 2007-1 1-21
[3] Johnson, Steven (2001). Emergence: the connected lives of ants, brains, cities, and software. New York: Scribner. p. 46.

ISBN 0-684-86875-X.
[4] Jacobs, Jane (1961). The Death and Life of Great American Cities. New York: Random House.
[5] Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997
[6] Johnson, Neil F. (2007). Two's Company, Three is Complexity: A simple guide to the science of all sciences. Oxford: Oneworld.

ISBN 978-1-85168-488-5.
[7] Lissack, Michael R.; Johan Roos (2000). The Next Common Sense, The e- Manager's Guide to Mastering Complexity. Intercultural Press.

ISBN 9781857882353.
[8] (http://www.constructal.Org/en/art/Phil.Trans.R.Soc.B (2010) 365, 1335D1347.pdf) Bejan A., Lorente S., The Constructal Law of

Design and Evolution in Nature. Philosophical Transactions of the Royal Society B, Biological Science, Vol. 365, 2010, pp. 1335-1347.
[9] Lorente S., Bejan A. (2010). Few Large and Many Small: Hierarchy in Movement on Earth, International Journal of Design of Nature and

Ecodynamics, Vol. 5, No. 3, pp. 254-267.
[10] Kim S., Lorente S., Bejan A., Milter W., Morse J. (2008) The Emergence of Vascular Design in Three Dimensions, Journal of Applied

Physics, Vol. 103, 123511.

Further reading

Chu, Dominique (2011). Complexity: Against Systems. Springer. PMID 21287293.

Waldrop, M. Mitchell (1992). Complexity: The Emerging Science at the Edge of Order and Chaos. New York:

Simon & Schuster. ISBN 9780671767891.

Czerwinski, Tom; David Alberts (1997). Complexity, Global Politics, and National Security (http://www.

dodccrp.org/files/Alberts_Complexity_Global.pdf). National Defense University. ISBN 9781579060466.

Czerwinski, Tom (1998). Coping with the Bounds: Speculations on Nonlinearity in Military Affairs (http://www.

dodccrp.org/files/Czerwinski_Coping.pdf). CCRP. ISBN 9781414503158 (from Pavilion Press, 2004).

Lissack, Michael R.; Johan Roos (2000). The Next Common Sense, The e-Manager's Guide to Mastering

Complexity. Intercultural Press. ISBN 9781857882353.

Sole, R. V.; B. C. Goodwin (2002). Signs of Life: How Complexity Pervades Biology. Basic Books.

ISBN 9780465019281.

Moffat, James (2003). Complexity Theory and Network Centric Warfare (http://www.dodccrp.org/files/

Moffat_Complexity.pdf). CCRP. ISBN 9781893723115.

Smith, Edward (2006). Complexity, Networking, and Effects Based Approaches to Operations (http://www.

dodccrp.org/files/Smith_Complexity.pdf). CCRP. ISBN 9781893723184.

Heylighen, Francis (2008). " Complexity and Self-Organization (http://pespmcl.vub.ac.be/Papers/

ELIS-Complexity.pdf)". In Bates, Marcia J.; Maack, Mary Niles. Encyclopedia of Library and Information

Sciences. CRC. ISBN 9780849397127

Greenlaw, N. and Hoover, H.J. Fundamentals of the Theory of Computation, Morgan Kauffman Publishers, San

Francisco, 1998

Blum, M. (1967) On the Size of Machines, Information and Control, v. 1 1, pp. 257—265

Burgin, M. (1982) Generalized Kolmogorov complexity and duality in theory of computations, Notices of the

Russian Academy of Sciences, v. 25, No. 3, pp. 19—23

Mark Burgin (2005), Super-recursive algorithms, Monographs in computer science, Springer.

Burgin, M. and Debnath, N. Hardship of Program Utilization and User-Friendly Software, in Proceedings of the

International Conference "Computer Applications in Industry and Engineering" , Las Vegas, Nevada, 2003,

pp. 314-317

Complexity 407

• Debnath, N.C. and Burgin, M., (2003) Software Metrics from the Algorithmic Perspective, in Proceedings of the
ISCA 18th International Conference "Computers and their Applications" , Honolulu, Hawaii, pp. 279—282

• Meyers, R. A., (2009) "Encyclopedia of Complexity and Systems Science", ISBN 978-0-387-75888-6

• Caterina Liberati, J. Andrew Howe, Hamparsum Bozdogan, Data Adaptive Simultaneous Parameter and Kernel
Selection in Kernel Discriminant Analysis Using Information Complexity (http://jprr.org/index.php/jprr/
article/view/ 1 17), Journal of Pattern Recognition Research, JPRR (http://www.jprr.org), Vol 4, No 1, 2009.

• Gershenson, C. and F. Heylighen (2005). How can we think the complex? (http://uk.arxiv.org/abs/nlin.AO/
0402023) In Richardson, Kurt (ed.) Managing Organizational Complexity: Philosophy, Theory and Application,
Chapter 3. Information Age Publishing.

External links

• Complexity Measures (http://cscs.umich.edu/~crshalizi/notebooks/complexity-measures.html) — an article
about the abundance of not-that-useful complexity measures.

• Exploring Complexity in Science and Technology (http://web.cecs.pdx.edu/~mm/
ExploringComplexityFall2009/index.html) — -ntroductory complex system course by Melanie Mitchell

• Quantifying Complexity Theory (http://www.calresco.org/lucas/quantify.htm) — classification of complex
systems

• Santa Fe Institute (http://www.santafe.edu/) focusing on the study of complexity science: Lecture Videos
(http://www.santafe.edu/research/videos/catalog/)

• UC Four Campus Complexity Videoconferences (http://eclectic.ss.uci.edu/~drwhite/center/cac.html) —
Human Sciences and Complexity

At the Edge of Chaos

The phrase edge of chaos was coined by mathematician Doyne Farmer to describe the transition phenomenon
discovered by computer scientist Christopher Langton. The phrase originally refers to an area in the range of a
variable, X (lambda), which was varied while examining the behavior of a cellular automaton (CA). As X varied, the
behavior of the CA went through a phase transition of behaviors. Langton found a small area conducive to produce
CAs capable of universal computation. At around the same time physicist James P. Crutchfield and others used the
phrase onset of chaos to describe more or less the same concept.

In the sciences in general, the phrase has come to refer to a metaphor that some physical, biological, economic and
social systems operate in a region between order and either complete randomness or chaos, where the complexity is
maximal. The generality and significance of the idea, however, has since been called into question by Melanie
Mitchell and others. The phrase has also been borrowed by the business community and is sometimes used
inappropriately and in contexts that are far from the original scope of the meaning of the term.

Stuart Kauffman has studied mathematical models of evolving systems in which the rate of evolution is maximized
near the edge of chaos.

At the Edge of Chaos 408

References

• Christopher G. Langton. "Computation at the edge of chaos". Physica D, 42, 1990.

• J. P. Crutchfield and K. Young, "Computation at the Onset of Chaos", in Entropy, Complexity, and the Physics of
Information, W. Zurek, editor, SFI Studies in the Sciences of Complexity, VIII, Addison-Wesley, Reading,
Massachusetts (1990) pp. 223-269.

• Melanie Mitchell, Peter T. Hraber, and James P. Crutchfield. Revisiting the edge of chaos: Evolving cellular
automata to perform computations . Complex Systems, 7:89—130, 1993.

• Melanie Mitchell, James P. Crutchfield and Peter T. Hraber. Dynamics, Computation, and the 'Edge of Chaos": A

m
Re-Examination

• Origins of Order: Self-Organization and Selection in Evolution by Stuart Kauffman

External links

• "The Edge of Chaos" - a criticism of the idea's prevalence.

References

[1] http://web.cecs.pdx.edu/~mm/rev-edge.pdf

[2] http://web.cecs.pdx.edu/~mm/dyn-comp-edge.pdf

[3] http://cscs.umich.edu/~crshalizi/notebooks/edge-of-chaos.html

Chaos control

In chaos theory, control of chaos is based on the fact that any chaotic attractor contains an infinite number of
unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves in the
neighborhood of one of these orbits for a while, then falls close to a different unstable periodic orbit where it remains
for a limited time, and so forth. This results in a complicated and unpredictable wandering over longer periods of
time.

Control of chaos is the stabilization, by means of small system perturbations, of one of these unstable periodic
orbits. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage.
The perturbation must be tiny, to avoid significant modification of the system's natural dynamics.

Several techniques have been devised for chaos control, but most are developments of two basic approaches: the
OGY (Ott, Grebogi and Yorke) method, and Pyragas continuous control. Both methods require a previous
determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.

In the OGY method, small, wisely chosen, swift kicks are applied to the system once per cycle, to maintain it near
the desired unstable periodic orbit. In the Pyragas method, an appropriate continuous controlling signal is injected
into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but
increases when it drifts away from the desired orbit.

Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including
turbulent fluids, oscillating chemical reactions, magneto-mechanical oscillators, and cardiac tissues. Sarnobat et al.
(2000) attempt the control of chaotic bubbling with the OGY method and using electrostatic potential as the primary
control variable.

The number of publications devoted to control of chaos is huge, see e.g. Chaos control bibliography (1997-2000)

Forcing two systems into the same state is not the only way to achieve synchronization of chaos. Both control of
chaos and synchronization constitute parts of Cybernetical Physics. Cybernetical physics is a research area on the
border between Physics and Control Theory.

Chaos control

409

References

• Sarnobat, S.U., "Modification, Identification & Control of Chaotic Bubbling via Electrostatic Potential", Masters

T21
Thesis, University of Tennessee, Knoxville, August 2000. link

• Cybernetical_physics

External links

• Chaos control bibliography ( 1 997-2000)

[l]

Books

Eckehard Scholl and Heinz Georg Schuster (Eds). Handbook of Chaos Control Wiley-VCH; 2nd Revision, Enlarged
edition (2007) Weinheim.

Gonzalez-Miranda, J. M. (2004). Synchronization and Control of Chaos: An Introduction for Scientists and
Engineers. London: Imperial College Press.

Fradkov A.L., Pogromsky A.Yu. (1998). Introduction to Control of Oscillations and Chaos. Singapore: World
Scientific Publishers.

References

[1] http://www.rusycon.ru/chaos-control.html
[2] http://www.chaoticbubbles.com

Butterfly effect

In chaos theory, the butterfly effect is the

The butterfly effect is a common trope in
fiction when presenting scenarios involving
time travel and with "what if" cases where
one storyline diverges at the moment of a

Sensitive dependency
on initial conditions

jf J k attractor D

Key: Blue squares represent initial states;
black circles represent equilibria

Point attractors in 2D phase space.

seemingly minor event resulting in two significantly different outcomes.

Butterfly effect 410

Theory

Origin of the concept and the term

as a shortcut on a number in the sequence, he entered the decimal .506 instead of entering the full .506127. The

T21 131

result was a completely different weather scenario. Lorenz published his findings in a 1963 paper for the New

York Academy of Sciences noting that "One meteorologist remarked that if the theory were correct, one flap of a

seagull's wings could change the course of weather forever." Later speeches and papers by Lorenz used the more

poetic butterfly. According to Lorenz, when Lorenz failed to provide a title for a talk he was to present at the 139th

meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the

flap of a butterfly's wings in Brazil set off a tornado in Texas? as a title. Although a butterfly flapping its wings has

remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location

of the consequences have varied widely.

Illustration

The butterfly effect in the Lorenz attractor

time < t < 30 (larger)

Z coordinate (larger)

Butterfly effect

411

A Java animation of the Lorenz attractor shows the continuous evolution.

Mathematical definition

A dynamical system with evolution map f * displays sensitive dependence on initial conditions if points arbitrarily
close together become separate with increasing t at an exponential rate. The definition is not topological, but
essentially metrical.

If M is the state space for the map f* , then f * displays sensitive dependence to initial conditions if for any x in M
and any 6>0, there are y in M, with < d(x, y) < S such that

d{f T (x)J T (y)) > exp(ar)d(x,y).
The definition does not require that all points from a neighborhood separate from the base point x, but it requires one
positive Lyapunov exponent.

Examples

The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction
models, for example

[5]

[6] [7]

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of
cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem
Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure
quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is

included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers.

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the
time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of

1121

sensitivity of quantum systems to small changes in their given Hamiltonians. Poulin et al. present a quantum
algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected
to slightly different dynamics." They consider fidelity decay to be "the closest quantum analog to the (purely
classical) butterfly effect." Whereas the classical butterfly effect considers the effect of a small change in the
position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect
of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly

Butterfly effect 412

effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to
initial conditions are known as quantum chaos.

References

[3] Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow" (http://journals.ametsoc.org/doi/abs/10.1175/

1520-0469(1963)020<0130:DNF>2.0.CO;2). Journal of the Atmospheric Sciences 20 (2): 130-141.

doi:10.1063/l. 1665596.

[12] Karkuszewski, Zbyszek P.; Jarzynski, Christopher; Zurek, Wojciech H. (2002). "Quantum Chaotic Environments, the Butterfly Effect, and

Information: Measuring the Average Fidelity Decay". Physical Review Letters 92 (17): 177906. Bibcode 2004PhRvL..92q7906P.

3949. doi: 10. 1063/1. 1788661.

Further reading

• Devaney, Robert L. (2003). Introduction to Chaotic Dynamical Systems. Westview Press. ISBN 0813340853.

External links

• From butterfly wings to single e-mail (http://www.news.cornell.edu/releases/Feb04/AAAS.Kleinberg.ws.
html) (Cornell University)

• New England Complex Systems Institute - Concepts: Butterfly Effect (http://necsi.edu/guide/concepts/
butterflyeffect. html)

• The Chaos Hypertextbook (http://hypertextbook.com/chaos/). An introductory primer on chaos and fractals.

• Weisstein, Eric W., " Butterfly Effect (http://mathworld.wolfram.com/ButterflyEffect.html)" from
MathWorld.

413

Applications

Data storage

Data storage can refer to:

• Computer data storage; memory, components, devices and media that retain digital computer data used for
computing for some interval of time.

• Any data storage device; that records (stores) or retrieves (reads) information (data) from any medium, including
the medium itself.

Online data storage solutions

Today, more and more people are mobile: they are working on similar tasks at home, office, etc.. Thus, the concept
of storing data online is becoming increasingly common. ContactOffice offers its users through its virtual office a
document storage space accessible online via their collaboration suite's Web interface, but also via the WebDAV
protocol as a network folder on the user's computer.

Data transmission

Data transmission, digital transmission, or digital communications is the physical transfer of data (a digital bit
stream) over a point-to-point or point-to-multipoint communication channel. Examples of such channels are copper
wires, optical fibres, wireless communication channels, and storage media. The data is represented as an
electromagnetic signal, such as an electrical voltage, radiowave, microwave, or infrared signal.

While analog communications is the transfer of continuously varying information signal, digital communications is
the transfer of discrete messages. The messages are either represented by a sequence of pulses by means of a line
code (baseband transmission), or by a limited set of continuously varying wave forms (passband transmission),
using a digital modulation method. The passband modulation and corresponding demodulation (also known as
detection) is carried out by modem equipment. According to the most common definition of digital signal, both
baseband and passband signals representing bit-streams are considered as digital transmission, while an alternative
definition only considers the baseband signal as digital, and passband transmission of digital data as a form of
digital-to-analog conversion.

Data transmitted may be digital messages originating from a data source, for example a computer or a keyboard. It
may also be an analog signal such as a phone call or a video signal, digitized into a bit-stream for example using
pulse-code modulation (PCM) or more advanced source coding (analog-to-digital conversion and data compression)
schemes. This source coding and decoding is carried out by codec equipment.

Data transmission 414

Distinction between related subjects

Courses and textbooks in the field of data transmission as well as digital transmission and digital

communications have similar content.

Digital transmission or data transmission traditionally belongs to telecommunications and electrical engineering.
Basic principles of data transmission may also be covered within the computer science/computer engineering topic
of data communications, which also includes computer networking or computer communication applications and
networking protocols, for example routing, switching and process-to-process communication. Although the
Transmission control protocol (TCP) involves the term "transmission", TCP and other transport layer protocols are
typically not discussed in a textbook or course about data transmission, but in computer networking.

The term tele transmission involves the analog as well as digital communication. In most textbooks, the term analog
transmission only refers to the transmission of an analog message signal (without digitization) by means of an analog
signal, either as a non-modulated baseband signal, or as a passband signal using an analog modulation method such
as AM or FM. It may also include analog-over-analog pulse modulatated baseband signals such as pulse-width
modulation. In a few books within the computer networking tradition, "analog transmission" also refers to passband
transmission of bit-streams using digital modulation methods such as FSK, PSK and ASK. Note that these methods
are covered in textbooks named digital transmission or data transmission, for example.

The theoretical aspects of data transmission are covered by information theory and coding theory.

Protocol layers and sub -topics

Courses and textbooks in the field of data transmission typically deal with the following OSI model protocol layers
and topics:

• Layer 1, the physical layer:

• Channel coding including

• Digital modulation schemes

• Line coding schemes

• Forward error correction (FEC) codes
Bit synchronization
Multiplexing
Equalization
Channel models

Layer 2, the data link layer:

Channel access schemes, media access control (MAC)
Packet mode communication and Frame synchronization
Error detection and automatic repeat request (ARQ)
Flow control
Layer 6, the presentation layer:

Source coding (digitization and data compression), and information theory.
Cryptography (may occur at any layer)

Data transmission 415

Applications and history

Data (mainly but not exclusively informational) has been sent via non-electronic (e.g. optical, acoustic, mechanical)
means since the advent of communication. Analog signal data has been sent electronically since the advent of the
telephone. However, the first data electromagnetic transmission applications in modern time were telegraphy (1809)
and teletypewriters (1906), which are both digital signals. The fundamental theoretical work in data transmission and
information theory by Harry Nyquist, Ralph Hartley, Claude Shannon and others during the early 20th century, was
done with these applications in mind.

Data transmission is utilized in computers in computer buses and for communication with peripheral equipment via
parallel ports and serial ports such us RS-232 (1969), Firewire (1995) and USB (1996). The principles of data
transmission is also utilized in storage media for Error detection and correction since 1951.

Data transmission is utilized in computer networking equipment such as modems (1940), local area networks (LAN)
adapters (1964), repeaters, hubs, microwave links, wireless network access points (1997), etc.

In telephone networks, digital communication is utilized for transferring many phone calls over the same copper
cable or fiber cable by means of Pulse code modulation (PCM), i.e. sampling and digitization, in combination with
Time division multiplexing (TDM) (1962). Telephone exchanges have become digital and software controlled,
facilitating many value added services. For example the first AXE telephone exchange was presented in 1976. Since
late 1980th, digital communication to the end user has been possible using Integrated Services Digital Network
(ISDN) services. Since the end of 1990th, broadband access techniques such as ADSL, Cable modems,
fiber-to-the-building (FTTB) and fiber-to-the-home (FTTH) have become wide spread to small offices and homes.
The current tendency is to replace traditional telecommunication services by packet mode communication such as IP
telephony and IPTV.

Transmitting analog signals digitally allows for greater signal processing capability. The ability to process a
communications signal means that errors caused by random processes can be detected and corrected. Digital signals
can also be sampled instead of continuously monitored. The multiplexing of multiple digital signals is much simpler
to the multiplexing of analog signals.

Because of all these advantages, and because recent advances in wideband communication channels and solid-state
electronics have allowed scientists to fully realize these advantages, digital communications has grown quickly.
Digital communications is quickly edging out analog communication because of the vast demand to transmit
computer data and the ability of digital communications to do so.

The digital revolution has also resulted in many digital telecommunication applications where the principles of data
transmission are applied. Examples are second-generation (1991) and later cellular telephony, video conferencing,
digital TV (1998), digital radio (1999), telemetry, etc.

Baseband or passband transmission

The physically transmitted signal may be one of the following:

1. A baseband signal ("digital-over-digital" transmission): A sequence of electrical pulses or light pulses produced
by means of a line coding scheme such as Manchester coding. This is typically used in serial cables, wired local
area networks such as Ethernet, and in optical fiber communication. It results in a pulse amplitude modulated
signal, also known as a pulse train.

2. A passband signal ("digital-over-analog" transmission): A modulated sine wave signal representing a digital
bit-stream. Note that this is in some textbooks considered as analog transmission, but in most books as digital
transmission. The signal is produced by means of a digital modulation method such as PSK, QAM or FSK. The
modulation and demodulation is carried out by modem equipment. This is used in wireless communication, and
over telephone network local-loop and cable-TV networks.

Data transmission 416

Serial and parallel transmission

In telecommunications, serial transmission is the sequential transmission of signal elements of a group representing a
character or other entity of data. Digital serial transmissions are bits sent over a single wire, frequency or optical path
sequentially. Because it requires less signal processing and less chances for error than parallel transmission, the
transfer rate of each individual path may be faster. This can be used over longer distances as a check digit or parity
bit can be sent along it easily.

In telecommunications, parallel transmission is the simultaneous transmission of the signal elements of a character or
other entity of data. In digital communications, parallel transmission is the simultaneous transmission of related
signal elements over two or more separate paths. Multiple electrical wires are used which can transmit multiple bits
simultaneously, which allows for higher data transfer rates than can be achieved with serial transmission. This
method is used internally within the computer, for example the internal buses, and sometimes externally for such
things as printers, The major issue with this is "skewing" because the wires in parallel data transmission have slightly
different properties (not intentionally) so some bits may arrive before others, which may corrupt the message. A
parity bit can help to reduce this. However, electrical wire parallel data transmission is therefore less reliable for long
distances because corrupt transmissions are far more likely.

Types of communication channels

• Simplex

• Half-duplex

• Full-duplex

• Point-to-point

• Multi-drop:

• Bus network

• Ring network

• Star network

• Mesh network

• Wireless network

Asynchronous and synchronous data transmission

Asynchronous transmission uses start and stop bits to signify the beginning bit ASCII character would actually be
transmitted using 10 bits e.g.: A "0100 0001" would become "1 0100 0001 0". The extra one (or zero depending on
parity bit) at the start and end of the transmission tells the receiver first that a character is coming and secondly that
the character has ended. This method of transmission is used when data is sent intermittently as opposed to in a solid
stream. In the previous example the start and stop bits are in bold. The start and stop bits must be of opposite
polarity. This allows the receiver to recognize when the second packet of information is being sent.

Synchronous transmission uses no start and stop bits but instead synchronizes transmission speeds at both the
receiving and sending end of the transmission using clock signal(s) built into each component. A continual stream of
data is then sent between the two nodes. Due to there being no start and stop bits the data transfer rate is quicker
although more errors will occur, as the clocks will eventually get out of sync, and the receiving device would have
the wrong time that had been agreed in the protocol for sending/receiving data, so some bytes could become
corrupted (by losing bits). Ways to get around this problem include re-synchronization of the clocks and use of check
digits to ensure the byte is correctly interpreted and received

Data transmission 417

Notes

[1] A. P. Clark , "Principles of Digital Data Transmission", Published by Wiley, 1983

[2] David R. Smith, "Digital Transmission Systems", Kluwer International Publishers, 2003, ISBN 1-4020-7587-1. See table-of-contents (http://

www.amazon.com/dp/ 140207587 1).
[3] Sergio Benedetto, Ezio Biglieri, "Principles of Digital Transmission: With Wireless Applications", Springer 2008, ISBN 0-306-45753-9,

9780306457531. See table-of-contents (http://search.barnesandnoble.com/Principles-of-Digital-Transmission-with-Wireless-Applications/

Sergio-Benedetto/e/9780306457531#TOC)
[4] Simon Haykin, "Digital Communications", John Wiley & Sons, 1988. ISBN 978-0-471-62947-4. See table-of-contents (http://www.

amazon.eom/dp/0471432229#reader_0471432229).
[5] John Proakis, "Digital Communications", 4th edition, McGraw-Hill, 2000. ISBN 0-07-2321 1 1-3. See table-of-contents (http://www.mhhe.

com/engcs/electrical/proakis/toc.mhtml).

External links

• Asynchronous serial data example (http://halowave.webs.com)
ltg:Datu puorlaida

418

Related Biographies

Isaac Newton

Sir Isaac Newton

Godfrey Kneller's 1689 portrait of Isaac Newton
(age 46)

Born

,[1],

4 January 1643

[OS: 25 December 1642] L * J Woolsthorpe-by-Colsterworth

Lincolnshire, England

Died

Residence

31 March 1727 (aged 84)

[OS: 20 March 1726] Kensington, Middlesex, England

England

Nationality English

Fields physics, mathematics, astronomy, natural philosophy, alchemy, Christian theology

Institutions

Alma mater

University of Cambridge
Royal Society
Royal Mint

Trinity College, Cambridge

Academic advisors

Isaac Barrow

r-\

Benjamin Pulleyn

[3] [4]

Notable students Roger Cotes

William Whiston

Known for Newtonian mechanics

Universal gravitation
Infinitesimal calculus
Optics

Binomial series
Newton's method
Philosophise Naturalis Principia Mathematica

Influences

[5]

Henry More
Polish Brethren 1

[6]

Influenced

Nicolas Fatio de Duillier
John Keill

Isaac Newton 419

Signature

Notes

His mother was Hannah Ayscough. His half-niece was Catherine Barton.

Sir Isaac Newton PRS (4 January 1643 - 31 March 1727 [OS: 25 December 1642 - 20 March 1727]) [1] was an English
physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian. His monograph Philosophic?
Naturalis Principia Mathematica, published in 1687, lays the foundations for most of classical mechanics and is one
of the most important scientific books ever written. In this work, Newton described universal gravitation and the
three laws of motion, which dominated the scientific view of the physical universe for the next three centuries.
Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural
laws, by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation; thus
removing the last doubts about heliocentrism and advancing the Scientific Revolution.

171
Newton built the first practical reflecting telescope and developed a theory of colour based on the observation that

a prism decomposes white light into the many colours that form the visible spectrum. He also formulated an

empirical law of cooling and studied the speed of sound.

In mathematics, Newton shares the credit with Gottfried Leibniz for the development of differential and integral
calculus. He also demonstrated the generalised binomial theorem, developed Newton's method for approximating the
roots of a function, and contributed to the study of power series.

Newton was also highly religious. He was an unorthodox Christian, and during his lifetime actually wrote more on
Biblical hermeneutics and occult studies than on science and mathematics, the subjects he is mainly associated with.
Newton secretly rejected Trinitarianism, fearing to be accused of refusing holy orders. Later studies on his biography

ro]

claimed that he was influenced by Muslim Arab Scholars, too.

Newton is considered by many scholars and members of the general public to be one of the most influential people
in human history.

Life
Early life

Isaac Newton was born on 4 January 1643 [OS: 25 December 1642] at Woolsthorpe Manor in
Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. At the time of Newton's birth, England had
not adopted the Gregorian calendar and therefore his date of birth was recorded as Christmas Day, 25 December
1642. Newton was born three months after the death of his father, a prosperous farmer also named Isaac Newton.
Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a
quart mug (a 1.1 litres). When Newton was three, his mother remarried and went to live with her new husband, the
Reverend Barnabus Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough. The young
Isaac disliked his stepfather and held some enmity towards his mother for marrying him, as revealed by this entry in

a list of sins committed up to the age of 19: "Threatening my father and mother Smith to burn them and the house

191
over them." While Newton was once engaged in his late teens to a Miss Storey, he never married, being highly

engrossed in his studies and work.

Isaac Newton

420

From the age of about twelve until he was seventeen, Newton was educated at
The King's School, Grantham (where his alleged signature can still be seen upon

ri3i

a library window sill) .He was removed from school, and by October 1659,
he was to be found at Woolsthorpe-by-Colsterworth, where his mother, widowed
by now for a second time, attempted to make a farmer of him. He hated

ri4i

farming. Henry Stokes, master at the King's School, persuaded his mother to
send him back to school so that he might complete his education. Motivated
partly by a desire for revenge against a schoolyard bully, he became the

top-ranked student

[15]

In June 1661, he was admitted to Trinity College, Cambridge as a sizar — a sort
of work-study role. At that time, the college's teachings were based on those
of Aristotle, but Newton preferred to read the more advanced ideas of modern
philosophers, such as Descartes, and of astronomers such as Copernicus, Galileo,
and Kepler. In 1665, he discovered the generalised binomial theorem and began
to develop a mathematical theory that would later become infinitesimal calculus.
Soon after Newton had obtained his degree in August 1665, the university
temporarily closed as a precaution against the Great Plague. Although he had

ri7i

been undistinguished as a Cambridge student, Newton's private studies at his
home in Woolsthorpe over the subsequent two years saw the development of his

theories on calculus, optics and the law of gravitation. In 1667, he returned to

n ri
Cambridge as a fellow of Trinity. Fellows were required to become ordained

priests, something Newton desired to avoid due to his unorthodox views. Luckily

for Newton, there was no specific deadline for ordination and it could be

postponed indefinitely. The problem became more severe later when Newton

was elected for the prestigious Lucasian Chair. For such a significant

appointment, ordaining normally could not be dodged. Nevertheless, Newton

managed to avoid it by means of a special permission from Charles II (see

"Middle years" section below).

Newton in a 1702 portrait by
Godfrey Kneller

Isaac Newton (Bolton, Sarah K.

Famous Men of Science. NY: Thomas

Y. Crow ell & Co.. 1889)

Middle years

Mathematics

ri9i

Newton's work has been said "to distinctly advance every branch of mathematics then studied".

His work on the subject usually referred to as fluxions or calculus is seen, for example, in a manuscript of October
1666, now published among Newton's mathematical papers. A related subject was infinite series. Newton's
manuscript "De analysi per aequationes numero terminorum infinitas" ("On analysis by equations infinite in number
of terms") was sent by Isaac Barrow to John Collins in June 1669: in August 1669 Barrow identified its author to
Collins as "Mr Newton, a fellow of our College, and very young ... but of an extraordinary genius and proficiency in
these things".

Newton later became involved in a dispute with Leibniz over priority in the development of infinitesimal calculus.
Most modern historians believe that Newton and Leibniz developed infinitesimal calculus independently, although
with very different notations. Occasionally it has been suggested that Newton published almost nothing about it until
1693, and did not give a full account until 1704, while Leibniz began publishing a full account of his methods in
1684. (Leibniz's notation and "differential Method", nowadays recognised as much more convenient notations, were
adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.) Such a
suggestion, however, fails to notice the content of calculus which critics of Newton's time and modern times have

Isaac Newton 421

pointed out in Book 1 of Newton's Principia itself (published 1687) and in its forerunner manuscripts, such as De
motu corporum in gyrum ("On the motion of bodies in orbit"), of 1684. The Principia is not written in the language
of calculus either as we know it or as Newton's (later) 'dot' notation would write it. But his work extensively uses an

infinitesimal calculus in geometric form, based on limiting values of the ratios of vanishing small quantities: in the

T221
Principia itself Newton gave demonstration of this under the name of 'the method of first and last ratios and

[231
explained why he put his expositions in this form, remarking also that 'hereby the same thing is performed as by

the method of indivisibles'.

Because of this, the Principia has been called "a book dense with the theory and application of the infinitesimal

T241
calculus" in modern times and "lequel est presque tout de ce calcul" ('nearly all of it is of this calculus') in

T251
Newton's time. His use of methods involving "one or more orders of the infinitesimally small" is present in his De

motu corporum in gyrum of 1684 and in his papers on motion "during the two decades preceding 1684".

no]

Newton had been reluctant to publish his calculus because he feared controversy and criticism. He had a very

close relationship with Swiss mathematician Nicolas Fatio de Duillier, who from the beginning was impressed by

Newton's gravitational theory. In 1691, Duillier planned to prepare a new version of Newton's Principia, but never

finished it. However, in 1693 the relationship between the two men changed. At the time, Duillier had also

[291
exchanged several letters with Leibniz.

Starting in 1699, other members of the Royal Society (of which Newton was a member) accused Leibniz of
plagiarism, and the dispute broke out in full force in 1711. The Royal Society proclaimed in a study that it was
Newton who was the true discoverer and labelled Leibniz a fraud. This study was cast into doubt when it was later
found that Newton himself wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy
which marred the lives of both Newton and Leibniz until the latter's death in 1716.

Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's
identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made
substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ
coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic
series by logarithms (a precursor to Euler's summation formula), and was the first to use power series with
confidence and to revert power series.

He was appointed Lucasian Professor of Mathematics in 1669 on Barrow's recommendation. In that day, any fellow
of Cambridge or Oxford was required to become an ordained Anglican priest. However, the terms of the Lucasian
professorship required that the holder not be active in the church (presumably so as to have more time for science).
Newton argued that this should exempt him from the ordination requirement, and Charles II, whose permission was
needed, accepted this argument. Thus a conflict between Newton's religious views and Anglican orthodoxy was
averted.

Isaac Newton

422

Optics

From 1670 to 1672, Newton lectured on optics. During this period he
investigated the refraction of light, demonstrating that a prism could
decompose white light into a spectrum of colours, and that a lens and a
second prism could recompose the multicoloured spectrum into white
light. [33]

He also showed that the coloured light does not change its properties
by separating out a coloured beam and shining it on various objects.
Newton noted that regardless of whether it was reflected or scattered or
transmitted, it stayed the same colour. Thus, he observed that colour is
the result of objects interacting with already-coloured light rather than
objects generating the colour themselves. This is known as Newton's
theory of colour.

A replica of Newton's second Reflecting
telescope that he presented to the Royal Society
in 1672

From this work, he concluded that the lens of any refracting telescope

would suffer from the dispersion of light into colours (chromatic aberration). As a proof of the concept, he

1351
constructed a telescope using a mirror as the objective to bypass that problem. Building the design, the first

1351

known functional reflecting telescope, today known as a Newtonian telescope, involved solving the problem of a
suitable mirror material and shaping technique. Newton ground his own mirrors out of a custom composition of
highly reflective speculum metal, using Newton's rings to judge the quality of the optics for his telescopes. In late
1668 he was able to produce this first reflecting telescope. In 1671, the Royal Society asked for a demonstration
of his reflecting telescope. Their interest encouraged him to publish his notes On Colour, which he later expanded
into his Opticks. When Robert Hooke criticised some of Newton's ideas, Newton was so offended that he withdrew
from public debate. Newton and Hooke had brief exchanges in 1679-80, when Hooke, appointed to manage the
Royal Society's correspondence, opened up a correspondence intended to elicit contributions from Newton to Royal

no]

Society transactions, which had the effect of stimulating Newton to work out a proof that the elliptical form of
planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see
Newton's law of universal gravitation - History and De motu corporum in gyrum). But the two men remained
generally on poor terms until Hooke's death

[39]

Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser
medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films
(Opticks Bk.II, Props. 12), but still retained his theory of 'fits' that disposed corpuscles to be reflected or transmitted
(Props. 13). Later physicists instead favoured a purely wavelike explanation of light to account for the interference
patterns, and the general phenomenon of diffraction. Today's quantum mechanics, photons and the idea of
wave— particle duality bear only a minor resemblance to Newton's understanding of light.

In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles.
The contact with the theosophist Henry More, revived his interest in alchemy. He replaced the ether with occult
forces based on Hermetic ideas of attraction and repulsion between particles. John Maynard Keynes, who acquired
many of Newton's writings on alchemy, stated that "Newton was not the first of the age of reason: He was the last of
the magicians." Newton's interest in alchemy cannot be isolated from his contributions to science; however, he
did apparently abandon his alchemical researches. (This was at a time when there was no clear distinction between
alchemy and science.) Had he not relied on the occult idea of action at a distance, across a vacuum, he might not
have developed his theory of gravity. (See also Isaac Newton's occult studies.)

In 1704, Newton published Opticks, in which he expounded his corpuscular theory of light. He considered light to be
made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that
through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ...and may

Isaac Newton

423

T411
not Bodies receive much of their Activity from the Particles of Light which enter their Composition?" Newton

also constructed a primitive form of a frictional electrostatic generator, using a glass globe (Optics, 8th Query).

T421
In an article entitled "Newton, prisms, and the 'opticks' of tunable lasers it is indicated that Newton in his book

Opticks was the first to show a diagram using a prism as a beam expander. In the same book he describes, via

diagrams, the use of multiple-prism arrays. Some 278 years after Newton's discussion, multiple-prism expanders

became central to the development of narrow-linewidth tunable lasers. Also, the use of these prismatic beam

T421
expanders led to the multiple-prism dispersion theory.

Mechanics and gravitation

In 1679, Newton returned to his work on (celestial) mechanics, i.e.,
gravitation and its effect on the orbits of planets, with reference to
Kepler's laws of planetary motion. This followed stimulation by a brief
exchange of letters in 1679-80 with Hooke, who had been appointed to
manage the Royal Society's correspondence, and who opened a
correspondence intended to elicit contributions from Newton to Royal
Society transactions. Newton's reawakening interest in astronomical
matters received further stimulus by the appearance of a comet in the

winter of 1680-1681, on which he corresponded with John

T431
Flamsteed. After the exchanges with Hooke, Newton worked out a

proof that the elliptical form of planetary orbits would result from a

centripetal force inversely proportional to the square of the radius vector (see Newton's law of universal gravitation -

History and De motu corporum in gyrum). Newton communicated his results to Edmond Halley and to the Royal

Society in De motu corporum in gyrum, a tract written on about 9 sheets which was copied into the Royal Society's

Newton's own copy of his Principia, with
hand-written corrections for the second edition

[44]

This tract contained the nucleus that Newton developed and expanded to form

The Principia was published on 5 July 1687 with encouragement and financial help from Edmond Halley. In this
work, Newton stated the three universal laws of motion that enabled many of the advances of the Industrial
Revolution which soon followed and were not to be improved upon for more than 200 years, and are still the
underpinnings of the non-relativistic technologies of the modern world. He used the Latin word gravitas (weight) for
the effect that would become known as gravity, and defined the law of universal gravitation.

In the same work, Newton presented a calculus-like method of geometrical analysis by 'first and last ratios', gave the
first analytical determination (based on Boyle's law) of the speed of sound in air, inferred the oblateness of the
spheroidal figure of the Earth, accounted for the precession of the equinoxes as a result of the Moon's gravitational
attraction on the Earth's oblateness, initiated the gravitational study of the irregularities in the motion of the moon,
provided a theory for the determination of the orbits of comets, and much more.

Newton made clear his heliocentric view of the solar system — developed in a somewhat modern way, because

[45]
already in the mid- 1680s he recognised the "deviation of the Sun" from the centre of gravity of the solar system.

For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather

"the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World",

and this centre of gravity "either is at rest or moves uniformly forward in a right line" (Newton adopted the "at rest"

alternative in view of common consent that the centre, wherever it was, was at rest).

Newton's postulate of an invisible force able to act over vast distances led to him being criticised for introducing
"occult agencies" into science. Later, in the second edition of the Principia (1713), Newton firmly rejected such
criticisms in a concluding General Scholium, writing that it was enough that the phenomena implied a gravitational
attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame
hypotheses of things that were not implied by the phenomena. (Here Newton used what became his famous

Isaac Newton

424

expression Hypotheses nonfingo).

[48]

With the Principia, Newton became internationally recognised. He acquired a circle of admirers, including the

Swiss-born mathematician Nicolas Fatio de Duillier, with whom he formed an intense relationship that lasted until

[49]
1693, when it abruptly ended, at the same time that Newton suffered a nervous breakdown.

^F\

■

^PWi *

Later life

In the 1690s, Newton wrote a number of religious tracts dealing with the literal
interpretation of the Bible. Henry More's belief in the Universe and rejection of
Cartesian dualism may have influenced Newton's religious ideas. A manuscript
he sent to John Locke in which he disputed the existence of the Trinity was never
published. Later works — The Chronology of Ancient Kingdoms Amended (1728)
and Observations Upon the Prophecies of Daniel and the Apocalypse of St. John
(1733) — were published after his death. He also devoted a great deal of time to
alchemy (see above).

Newton was also a member of the Parliament of England from 1689 to 1690 and
in 1701, but according to some accounts his only comments were to complain
about a cold draught in the chamber and request that the window be closed.

Newton moved to London to take up the post of warden of the Royal Mint in
1696, a position that he had obtained through the patronage of Charles Montagu,
1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of
England's great recoining, somewhat treading on the toes of Lord Lucas,
Governor of the Tower (and securing the job of deputy comptroller of the
temporary Chester branch for Edmond Halley). Newton became perhaps the
best-known Master of the Mint upon the death of Thomas Neale in 1699, a
position Newton held until his death. These appointments were intended as
sinecures, but Newton took them seriously, retiring from his Cambridge duties in
1701, and exercising his power to reform the currency and punish clippers and
counterfeiters. As Master of the Mint in 1717 in the "Law of Queen Anne"
Newton moved the Pound Sterling de facto from the silver standard to the gold
standard by setting the bimetallic relationship between gold coins and the silver
penny in favour of gold. This caused silver sterling coin to be melted and shipped
out of Britain. Newton was made President of the Royal Society in 1703 and an
associate of the French Academie des Sciences. In his position at the Royal
Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by
prematurely publishing Flamsteed's Historia Coelestis Britannica, which Newton had used in his studies

Isaac Newton in old age in 1712,
portrait by Sir James Thornhill

Personal coat of arms of Sir Isaac
Newton

[52]

In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. The knighthood is
likely to have been motivated by political considerations connected with the Parliamentary election in May 1705,

[53]

rather than any recognition of Newton's scientific work or services as Master of the Mint. Newton was the second
scientist to be knighted, after Sir Francis Bacon.

Towards the end of his life, Newton took up residence at Cranbury Park, near Winchester with his niece and her
husband, until his death in 1727. Newton died in his sleep in London on 31 March 1727 [OS: 20 March 1726], and
was buried in Westminster Abbey. His half-niece, Catherine Barton Conduitt, served as his hostess in social
affairs at his house on Jermyn Street in London; he was her "very loving Uncle," according to his letter to her
when she was recovering from smallpox. Newton, a bachelor, had divested much of his estate to relatives during his
last years, and died intestate.

Isaac Newton 425

After his death, Newton's body was discovered to have had massive amounts of mercury in it, probably resulting

[571
from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life.

After death

Fame

French mathematician Joseph-Louis Lagrange often said that Newton was the greatest genius who ever lived, and
once added that Newton was also "the most fortunate, for we cannot find more than once a system of the world to

KOI

establish." English poet Alexander Pope was moved by Newton's accomplishments to write the famous epitaph:

Nature and nature's laws lay hid in night;
God said "Let Newton be" and all was light.

Newton himself had been rather more modest of his own achievements, famously writing in a letter to Robert Hooke
in February 1676:

If I have seen further it is by standing on the shoulders of giants.

Two writers think that the above quote, written at a time when Newton and Hooke were in dispute over optical
discoveries, was an oblique attack on Hooke (said to have been short and hunchbacked), rather than — or in addition
to — a statement of modesty. On the other hand, the widely known proverb about standing on the shoulders of

giants published among others by 17th-century poet George Herbert (a former orator of the University of Cambridge
and fellow of Trinity College) in his Jacula Prudentum (1651), had as its main point that "a dwarf on a giant's
shoulders sees farther of the two", and so its effect as an analogy would place Newton himself rather than Hooke as
the 'dwarf.

In a later memoir, Newton wrote:

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing
on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than
ordinary, whilst the great ocean of truth lay all undiscovered before me.

Newton remains influential to scientists, as demonstrated by a 2005 survey of members of Britain's Royal Society
(formerly headed by Newton) asking who had the greater effect on the history of science, Newton or Albert Einstein.
Royal Society scientists deemed Newton to have made the greater overall contribution. In 1999, an opinion poll
of 100 of today's leading physicists voted Einstein the "greatest physicist ever;" with Newton the runner-up, while a
parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton. Charles Murray
quantitatively ranked great innovators in his book Human Accomplishment and found Newton to be the second most
important person in mathematics, to be one of the two most important persons in physics, and to be the most
important person in all of science combined.

Isaac Newton

426

Commemorations

Newton's monument (1731) can be seen in Westminster Abbey, at the north of
the entrance to the choir against the choir screen, near his tomb. It was executed
by the sculptor Michael Rysbrack (1694—1770) in white and grey marble with
design by the architect William Kent. The monument features a figure of Newton
reclining on top of a sarcophagus, his right elbow resting on several of his great
books and his left hand pointing to a scroll with a mathematical design. Above
him is a pyramid and a celestial globe showing the signs of the Zodiac and the
path of the comet of 1680. A relief panel depicts putti using instruments such as a
telescope and prism. The Latin inscription on the base translates as:

Here is buried Isaac Newton, Knight, who by a strength of mind

almost divine, and mathematical principles peculiarly his own,

explored the course and figures of the planets, the paths of comets,

the tides of the sea, the dissimilarities in rays of light, and, what no

other scholar has previously imagined, the properties of the colours

thus produced. Diligent, sagacious and faithful, in his expositions of

nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and

good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed

such and so great an ornament of the human race! He was born on 25 December 1642, and died on 20

March 1726/7. — Translation from G.L. Smyth, The Monuments and Genii of St. Paul's Cathedral, and

Newton statue on display at the

Oxford University Museum of

Natural History

of Westminster Abbey (1826), ii, 703-4

[66]

From 1978 until 1988, an image of Newton designed by Harry Ecclestone appeared on Series D £1 banknotes issued
by the Bank of England (the last £1 notes to be issued by the Bank of England). Newton was shown on the reverse of
the notes holding a book and accompanied by a telescope, a prism and a map of the Solar System

[67]

A statue of Isaac Newton, standing over an apple, can be seen at the Oxford University Museum of Natural History.

Religious views

According to most scholars, Newton was a monotheist who believed in biblical
prophecies but was Antitrinitarian. In Newton's eyes, worshipping Christ

as God was idolatry, to him the fundamental sin'. Historian Stephen D.
Snobelen says of Newton, "Isaac Newton was a heretic. But ... he never made a
public declaration of his private faith — which the orthodox would have deemed
extremely radical. He hid his faith so well that scholars are still unravelling his
personal beliefs." Snobelen concludes that Newton was at least a Socinian
sympathiser (he owned and had thoroughly read at least eight Socinian books),
possibly an Arian and almost certainly an antitrinitarian. In an age notable for
its religious intolerance, there are few public expressions of Newton's radical
views, most notably his refusal to take holy orders and his refusal, on his death

bed, to take the sacrament when it was offered to him

[6],

[6]

Newton's tomb in Westminster
Abbey

In a view disputed by Snobelen, T.C. Pfizenmaier argues that Newton held the
Arian view of the Trinity rather than the Western one held by Roman Catholics,

Anglicans, and most Protestants. In his own day, he was also accused of being a Rosicrucian (as were many in the

1711
Royal Society and in the court of Charles II).

Isaac Newton

427

Although the laws of motion and universal gravitation became Newton's best-known discoveries, he warned against
using them to view the Universe as a mere machine, as if akin to a great clock. He said, "Gravity explains the
motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that
is or can be done."

His scientific fame notwithstanding, Newton's studies of the Bible and of the early Church Fathers were also
noteworthy. Newton wrote works on textual criticism, most notably An Historical Account of Two Notable
Corruptions of Scripture. He also placed the crucifixion of Jesus Christ at 3 April, AD 33, which agrees with one
traditionally accepted date. He also tried, unsuccessfully, to find hidden messages within the Bible.

Newton wrote more on religion than he did on natural science. He believed in a rationally immanent world, but he

rejected the hylozoism implicit in Leibniz and Baruch Spinoza. Thus, the ordered and dynamically informed

Universe could be understood, and must be understood, by an active reason. In his correspondence, Newton claimed

that in writing the Principia "I had an eye upon such Principles as might work with considering men for the belief of

a Deity". He saw evidence of design in the system of the world: "Such a wonderful uniformity in the planetary

system must be allowed the effect of choice". But Newton insisted that divine intervention would eventually be

[751
required to reform the system, due to the slow growth of instabilities. For this, Leibniz lampooned him: "God

Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems,

sufficient foresight to make it a perpetual motion
Samuel Clarke in a famous correspondence.

.,[76]

Newton's position was vigorously defended by his follower

Effect on religious thought

Newton and Robert Boyle's mechanical philosophy was promoted by rationalist pamphleteers as a viable alternative

to the pantheists and enthusiasts, and was accepted hesitantly by orthodox preachers as well as dissident preachers

T771
like the latitudinarians. Thus, the clarity and simplicity of science was seen as a way to combat the emotional and

T781

metaphysical superlatives of both superstitious enthusiasm and the threat of atheism, and, at the same time, the
second wave of English deists used Newton's discoveries to demonstrate the possibility of a "Natural Religion".

The attacks made against pre-Enlightenment "magical thinking", and

the mystical elements of Christianity, were given their foundation with

Boyle's mechanical conception of the Universe. Newton gave Boyle's

ideas their completion through mathematical proofs and, perhaps more

[79]
importantly, was very successful in popularising them. Newton

refashioned the world governed by an interventionist God into a world

crafted by a God that designs along rational and universal

principles. These principles were available for all people to

discover, allowed people to pursue their own aims fruitfully in this life,

not the next, and to perfect themselves with their own rational

[81]
powers.

"Newton", by William Blake; here, Newton is
depicted critically as a "divine geometer".

Newton saw God as the master creator whose existence could not be denied in the face of the grandeur of all

TS91 T831 TR41

creation. His spokesman, Clarke, rejected Leibniz' theodicy which cleared God from the responsibility for

I'origine du mal by making God removed from participation in his creation, since as Clarke pointed out, such a deity

roc]

would be a king in name only, and but one step away from atheism. But the unforeseen theological consequence
of the success of Newton's system over the next century was to reinforce the deist position advocated by Leibniz.
The understanding of the world was now brought down to the level of simple human reason, and humans, as Odo
Marquard argued, became responsible for the correction and elimination of evil

[87]

On the other hand, latitudinarian and Newtonian ideas taken too far resulted in the millenarians, a religious faction
dedicated to the concept of a mechanical Universe, but finding in it the same enthusiasm and mysticism that the

rpo]

Enlightenment had fought so hard to extinguish.

Isaac Newton 428

Views of the end of the world

In a manuscript he wrote in 1704 in which he describes his attempts to extract scientific information from the Bible,
he estimated that the world would end no earlier than 2060. In predicting this he said, "This I mention not to assert
when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently
predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions
fail." [89]

Enlightenment philosophers

Enlightenment philosophers chose a short history of scientific predecessors — Galileo, Boyle, and Newton
principally — as the guides and guarantors of their applications of the singular concept of Nature and Natural Law to
every physical and social field of the day. In this respect, the lessons of history and the social structures built upon it
could be discarded.

It was Newton's conception of the Universe based upon Natural and rationally understandable laws that became one
of the seeds for Enlightenment ideology. Locke and Voltaire applied concepts of Natural Law to political systems
advocating intrinsic rights; the physiocrats and Adam Smith applied Natural conceptions of psychology and
self-interest to economic systems; and sociologists criticised the current social order for trying to fit history into
Natural models of progress. Monboddo and Samuel Clarke resisted elements of Newton's work, but eventually
rationalised it to conform with their strong religious views of nature.

Counterfeiters

As warden of the Royal Mint, Newton estimated that 20 percent of the coins taken in during The Great Recoinage of

1696 were counterfeit. Counterfeiting was high treason, punishable by the felon's being hanged, drawn and

quartered. Despite this, convicting the most flagrant criminals could be extremely difficult. However, Newton

[92]
proved to be equal to the task. Disguised as a habitue of bars and taverns, he gathered much of that evidence

[93]
himself. For all the barriers placed to prosecution, and separating the branches of government, English law still

had ancient and formidable customs of authority. Newton had himself made a justice of the peace in all the home

counties. Then he conducted more than 100 cross-examinations of witnesses, informers, and suspects between June

[94]
1698 and Christmas 1699. Newton successfully prosecuted 28 coiners.

[95]
One of Newton's cases as the King's attorney was against William Chaloner. Chaloner's schemes included setting

up phony conspiracies of Catholics and then turning in the hapless conspirators whom he had entrapped. Chaloner

made himself rich enough to posture as a gentleman. Petitioning Parliament, Chaloner accused the Mint of providing

tools to counterfeiters (a charge also made by others). He proposed that he be allowed to inspect the Mint's processes

in order to improve them. He petitioned Parliament to adopt his plans for a coinage that could not be counterfeited,

while at the same time striking false coins. Newton put Chaloner on trial for counterfeiting and had him sent to

Newgate Prison in September 1697. But Chaloner had friends in high places, who helped him secure an acquittal and

[95]
his release. Newton put him on trial a second time with conclusive evidence. Chaloner was convicted of high

[97]
treason and hanged, drawn and quartered on 23 March 1699 at Tyburn gallows.

Isaac Newton 429

Laws of motion

The famous three laws of motion (stated in modernised form): Newton's First Law (also known as the Law of
Inertia) states that an object at rest tends to stay at rest and that an object in uniform motion tends to stay in uniform
motion unless acted upon by a net external force.

Newton's Second Law states that an applied force, ^, on an object equals the rate of change of its momentum, p ,
with time. Mathematically, this is expressed as

i! dp d i -» ^ dm dv

P — — - — —(mi;) — v — \-m— .

dt dV ' dt dt

If applied to an object with constant mass (dm/dt = 0), the first term vanishes, and by substitution using the definition
of acceleration, the equation can be written in the iconic form

F — ma .

The first and second laws represent a break with the physics of Aristotle, in which it was believed that a force was
necessary in order to maintain motion. They state that a force is only needed in order to change an object's state of
motion. The SI unit of force is the newton, named in Newton's honour.

Newton's Third Law states that for every action there is an equal and opposite reaction. This means that any force
exerted onto an object has a counterpart force that is exerted in the opposite direction back onto the first object. A
common example is of two ice skaters pushing against each other and sliding apart in opposite directions. Another
example is the recoil of a firearm, in which the force propelling the bullet is exerted equally back onto the gun and is
felt by the shooter. Since the objects in question do not necessarily have the same mass, the resulting acceleration of
the two objects can be different (as in the case of firearm recoil).

Unlike Aristotle's, Newton's physics is meant to be universal. For example, the second law applies both to a planet
and to a falling stone.

The vector nature of the second law addresses the geometrical relationship between the direction of the force and the
manner in which the object's momentum changes. Before Newton, it had typically been assumed that a planet
orbiting the sun would need a forward force to keep it moving. Newton showed instead that all that was needed was
an inward attraction from the sun. Even many decades after the publication of the Principia, this counterintuitive
idea was not universally accepted, and many scientists preferred Descartes' theory of vortices.

Apple analogy

Reputed descendants of Newton's apple tree, at the Cambridge University Botanic Garden and the Instituto Balseiro library garden

Isaac Newton 430

Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of
an apple from a tree.

Cartoons have gone further to suggest the apple actually hit Newton's head, and that its impact somehow made him
aware of the force of gravity, though this is not reported in the biographical manuscript by William Stukeley,
published in 1752, and made available by the Royal Society. It is known from his notebooks that Newton was
grappling in the late 1660s with the idea that terrestrial gravity extends, in an inverse-square proportion, to the
Moon; however it took him two decades to develop the full-fledged theory. John Conduitt, Newton's assistant at
the Royal Mint and husband of Newton's niece, described the event when he wrote about Newton's life:

In the year 1666 he retired again from Cambridge to his mother in Lincolnshire. Whilst he was
pensively meandering in a garden it came into his thought that the power of gravity (which brought an
apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must
extend much further than was usually thought. Why not as high as the Moon said he to himself & if so,
that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating what
would be the effect of that supposition.

The question was not whether gravity existed, but whether it extended so far from Earth that it could also be the
force holding the moon to its orbit. Newton showed that if the force decreased as the inverse square of the distance,
one could indeed calculate the Moon's orbital period, and get good agreement. He guessed the same force was
responsible for other orbital motions, and hence named it "universal gravitation".

Stukeley recorded in his Memoirs of Sir Isaac Newton's Life a conversation with Newton in Kensington on 15 April
1726, in which Newton recalled:

when formerly, the notion of gravitation came into his mind. It was occasioned by the fall of an apple, as
he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground,
thought he to himself. Why should it not go sideways or upwards, but constantly to the Earth's centre?
Assuredly the reason is, that the Earth draws it. There must be a drawing power in matter. And the sum
of the drawing power in the matter of the Earth must be in the Earth's centre, not in any side of the
Earth. Therefore does this apple fall perpendicularly or towards the centre? If matter thus draws matter;
it must be proportion of its quantity. Therefore the apple draws the Earth, as well as the Earth draws the
apple." [103]

In similar terms, Voltaire wrote in his Essay on Epic Poetry (1727), "Sir Isaac Newton walking in his gardens, had
the first thought of his system of gravitation, upon seeing an apple falling from a tree."

Various trees are claimed to be "the" apple tree which Newton describes. The King's School, Grantham, claims that
the tree was purchased by the school, uprooted and transported to the headmaster's garden some years later. The staff
of the [now] National Trust-owned Woolsthorpe Manor dispute this, and claim that a tree present in their gardens is
the one described by Newton. A descendant of the original tree can be seen growing outside the main gate of Trinity
College, Cambridge, below the room Newton lived in when he studied there. The National Fruit Collection at
Brogdale can supply grafts from their tree, which appears identical to Flower of Kent, a coarse-fleshed cooking

• t [105]

variety.

Isaac Newton 431

Writings

• Method of Fluxions (1671)

• Of Natures Obvious Laws & Processes in Vegetation (unpublished, c. 1671—75)

• De motu corporum in gyrum (1684)

• Philosophic Naturalis Principia Mathematica (1687)

• Opticks (1704)

• Reports as Master of the Mint [107] (1701-25)

• Arithmetica Universalis (1707)

• The System of the World, Optical Lectures, The Chronology of Ancient Kingdoms, (Amended) and De mundi
systemate (published posthumously in 1728)

• Observations on Daniel and The Apocalypse of St. John (1733)

• An Historical Account of Two Notable Corruptions of Scripture (1754)

Footnotes and references

[I] During Newton's lifetime, two calendars were in use in Europe: the Julian or 'Old Style' in Britain and parts of northern Europe (Protestant)
and eastern Europe, and the Gregorian or 'New Style', in use in Roman Catholic Europe and elsewhere. At Newton's birth, Gregorian dates
were ten days ahead of Julian dates: thus Newton was born on Christmas Day, 25 December 1642 by the Julian calendar, but on 4 January
1643 by the Gregorian. By the time he died, the difference between the calendars had increased to eleven days. Moreover, prior to the
adoption of the Gregorian calendar in the UK in 1752, the English new year began (for legal and some other civil purposes) on 25 March
('Lady Day', i.e. the feast of the Annunciation: sometimes called Annunciation Style') rather than on 1 January (sometimes called
'Circumcision Style'). Unless otherwise noted, the remainder of the dates in this article follow the Julian Calendar.

[2] Mordechai Feingold, Barrow, Isaac (1630—1677) (http://www.oxforddnb.com/view/article/1541), Oxford Dictionary of National

Biography, Oxford University Press, September 2004; online edn, May 2007; accessed 24 February 2009; explained further in Mordechai

Feingold " Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation (http://www.jstor.org/stable/236236)"; his, Vol. 84, No. 2

(June, 1993), pp. 310-338
[3] Dictionary of Scientific Biography (http://www.chlt.Org/sandbox/lhl/dsb/page.50.a.php), Newton, Isaac, n.4
[4] Gjersten, Derek (1986). The Newton Handbook. London: Routledge & Kegan Paul.
[5] Westfall, Richard S. (1983) [1980]. Never at Rest: A Biography of Isaac Newton. Cambridge: Cambridge University Press, pp. 530—1.

ISBN 9780521274357.
[6] Snobelen, Stephen D. (1999). "Isaac Newton, heretic: the strategies of a Nicodemite" (http://www.isaac-newton.org/heretic.pdf) (PDF).

British Journal for the History of Science 32 (4): 381-419. doi:10.1017/S0007087499003751. .
[7] "The Early Period (1608—1672)" (http://etoile.berkeley.edu/~jrg/TelescopeHistory/Early_Period.html). James R. Graham's Home Page. .

Retrieved 2009-02-03.
[8] Muslim Influences On Isaac Newton (http://www.daccatimes.com/?p=220)

[9] Cohen, LB. (1970). Dictionary of Scientific Biography, Vol. 11, p.43. New York: Charles Scribner's Sons
[10] "Isaac Newton's Life" (http://www.newton.ac.uk/newtlife.html). Isaac Newton Institute for Mathematical Sciences. 1998. . Retrieved

2010-03-28.

[II] "Isaac Newton" (http://scidiv.bellevuecollege.edu/MATH/Newton.html). Bellevue College. . Retrieved 2010-03-28.

[12] Newton, Isaac; Derek Thomas Whiteside (1967). The Mathematical Papers of Isaac Newton: 1664-1666 (http://books.google.com/

?id=lZcYsNBptfYC&pg=PA8&lpg=PA8&dq=isaac+newton+miss+storey&q=miss storey). Cambridge: Cambridge University Press.

p. 8. ISBN 9780521058179. . Retrieved 2010-03-28.
[13] http://www.flickr.com/photos/kingsschoollibrary/3645251382/in/photostream/
[14] Westfall 1994, pp 16-19
[15] White 1997, p. 22
[16] Michael White, Isaac Newton (1999) page 46 (http://books.google.com/books?id=12C3NV38tM0C&pg=PA24&dq=storer+

intitle:isaac+intitle:newton&lr=&num=30&as_brr=0&as_pt=ALLTYPES#PPA46,Ml)
[17] ed. Michael Hoskins (1997). Cambridge Illustrated History of Astronomy, p. 159. Cambridge University Press
[18] Newton, Isaac (http://venn.lib. cam. ac.uk/cgi-bin/search. pi ?sur=&suro=c&fir=&firo=c&cit=&cito=c&c=all&tex=RY644J&sye=&

eye=&col=all&maxcount=50) in Venn, J. & J. A., Alumni Cantabrigienses, Cambridge University Press, 10 vols, 1922—1958.
[19] W W Rouse Ball (1908), "A short account of the history of mathematics", at page 319.
[20] D T Whiteside (ed.), The Mathematical Papers of Isaac Newton (Volume 1), (Cambridge University Press, 1967), part 7 "The October 1666

Tract on Fluxions", at page 400, in 2008 reprint (http://books.google.com/books?id=lZcYsNBptfYC&pg=PA400).
[21] D Gjertsen (1986), "The Newton handbook", (London (Routledge & Kegan Paul) 1986), at page 149.
[22] Newton, 'Principia', 1729 English translation, at page 41 (http://books. google. com/books?id=TmOFAAAAQAAJ&pg=PA41).

Isaac Newton

432

[23] Newton, 'Principia', 1729 English translation, at page 54 (http://books. google. com/books?id=TmOFAAAAQAAJ&pg=PA54).

[24] Clifford Truesdell, Essays in the History of Mechanics (Berlin, 1968), at p.99.

[25] In the preface to the Marquis de L'Hospital's Analyse des lnfiniment Petits (Paris, 1696).

[26] Starting with De motu corporum in gyrum#Contents of 'De Motu', see also (Latin) Theorem 1 (http://books.google.com/

books?id=uvMGAAAAcAAJ&pg=RAl-PA2).
[27] D T Whiteside (1970), "The Mathematical principles underlying Newton's Principia Mathematica" in Journal for the History of Astronomy,

vol.1, pages 116-138, especially at pages 119-120.
[28] Stewart 2009, p. 107
[29] Westfall 1980, pp 538-539
[30] Ball 1908, p. 356ff
[31] White 1997, p. 151
[32] King, Henry C (2003). "The History of the Telescope" By Henry C. King, Page 74 (http://books.google.com/?id=KAWwzHlDVksC&

dq=history+of+the+telescope&printsec=frontcover). Books.google.com. ISBN 9780486432656. . Retrieved 2010-01-16.
[33] Ball 1908, p. 324
[34] Ball 1908, p. 325
[35] White 1997, pl70
[36] Hall, Alfred Rupert (1996). '"Isaac Newton: adventurer in thought'", by Alfred Rupert Hall, page 67 (http://books.google.com/

?id=32IDpTdthm4C&pg=PA67&lpg=PA67&dq=newton+reflecting+telescope++1668+letter+1669&q=newton reflecting telescope

1668 letter 1669). Books.google.com. ISBN 9780521566698. . Retrieved 2010-01-16.
[37] White 1997, pl68
[38] See 'Correspondence of Isaac Newton, vol.2, 1676-1687' ed. H W Turnbull, Cambridge University Press 1960; at page 297, document #235,

letter from Hooke to Newton dated 24 November 1679.
[39] Iliffe, Robert (2007) Newton. A very short introduction, Oxford University Press 2007
[40] Keynes, John Maynard (1972). "Newton, The Man". The Collected Writings of John Maynard Keynes Volume X. MacMillan St. Martin's

Press, pp. 363^.
[41] Dobbs, J.T. (December 1982). "Newton's Alchemy and His Theory of Matter". Isis 73 (4): 523. doi:10.1086/353114. quoting Opticks
[42] Duarte F. J (2000). " Newton, prisms, and the 'opticks' of tunable lasers (http://www.opticsjournal.com/FJ.DuarteOPN(2000).pdf)".

Optics and Photonics News 11 (5): 24-25. doi:10.1364/OPN.11.5.000024.
[43] R S Westfall, 'Never at Rest', 1980, at pages 391-2.

[44] D T Whiteside (ed.), 'Mathematical Papers of Isaac Newton', vol.6, 1684-1691, Cambridge University Press 1974, at page 30.
[45] See Curtis Wilson, "The Newtonian achievement in astronomy", pages 233-274 in R Taton & C Wilson (eds) (1989) The General History of

Astronomy, Volume, 2A', at page 233 (http://books.google.com/books ?id=rkQKU-wfPYMC&pg=PA233).
[46] Text quotations are from 1729 translation of Newton's Principia, Book 3 (1729 vol.2) at pages 232-233 (http://books.google.com/

books?id=6EqxPav3vIsC&pg=PA233).
[47] Edelglass et al., Matter and Mind, ISBN 0-940262-45-2. p. 54
[48] Westfall 1980. Chapter 11.

[49] Westfall 1980. pp 493^-97 on the friendship with Fatio, pp 531-540 on Newton's breakdown.
[50] Gerard Michon. "Coat of arms of Isaac Newton" (http://www.numericana.eom/arms/index.htm#newton). Numericana.com. . Retrieved

2010-01-16.
[51] White 1997, p. 232
[52] White 1997, p.3 17
[53] "The Queen's 'great Assistance' to Newton's election was his knighting, an honor bestowed not for his contributions to science, nor for his

service at the Mint, but for the greater glory of party politics in the election of 1705." Westfall 1994 p. 245
[54] Yonge, Charlotte M. (1898). "Cranbury and Brambridge" (http://www.online-literature.eom/charlotte-yonge/john-keble/6/). John

Keble's Parishes — Chapter 6. www.online-literature.com. . Retrieved 23 September 2009.
[55] Westfall 1980, p. 44.
[56] Westfall 1980, p. 595
[57] "Newton, Isaac (1642-1727)" (http://scienceworld.wolfram.com/biography/Newton.html). Eric Weisstein's World of Biography. .

Retrieved 2006-08-30.
[58] Fred L. Wilson, History of Science: Newton citing: Delambre, M. "Notice sur la vie et les ouvrages de M. le comte J. L. Lagrange," Oeuvres

de Lagrange I. Paris, 1867, p. xx.
[59] Letter from Isaac Newton to Robert Hooke, 5 February 1676, as transcribed in Jean-Pierre Maury (1992) Newton: Understanding the

Cosmos, New Horizons
[60] Wikipedia Standing on the shoulders of giants,
[61] John Gribbin (2002) Science: A History 1543-2001, p 164.
[62] White 1997, pl87.

[63] Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton (1855) by Sir David Brewster (Volume II. Ch. 27)
[64] "Newton beats Einstein in polls of Royal Society scientists and the public" (http://royalsociety. org/News. aspx?id=1324&

terms=Newton+beats+Einstein+in+polls+of+scientists+and+the+public). The Royal Society. .

Isaac Newton 433

[65] Opinion poll. Einstein voted "greatest physicist ever" by leading physicists; Newton runner-up: BBC news, Monday, 29 November 1999,

News.bbc.co.uk (http://news.bbc.co.Uk/2/hi/science/nature/541840.stm)
[66] "Famous People & the Abbey: Sir Isaac Newton" (http://www.westminster-abbey.org/our-history/people/sir-isaac-newton).

Westminster Abbey. . Retrieved 2009-11-13.
[67] "Withdrawn banknotes reference guide" (http://www.bankofengland.co.uk/banknotes/denom_guide/nonflash/l-SeriesD-Revised.htm).

Bank of England. . Retrieved 2009-08-27.
[68] Avery Cardinal Dulles. The Deist Minimum (http://www.firstthings.com/print.php?type=article&year=2008&month=08&

title_link=the-deist-minimum--28). January 2005.
[69] Westfall, Richard S. (1994). The Life of Isaac Newton. Cambridge: Cambridge University Press. ISBN 0521477379.
[70] Pfizenmaier, T.C. (1997). "Was Isaac Newton an Arian?". Journal of the History of Ideas 58 (1): 57—80.
[71] Yates, Frances A. (1972). The Rosicrucian Enlightenment. London: Routledge. ISBN 0415267692.

[72] Tiner, J.H. (1975). Isaac Newton: Inventor, Scientist and Teacher. Milford, Michigan, U.S.: Mott Media. ISBN 0915134950.
[73] John P. Meier, A Marginal Jew, v. 1, pp. 382—402 after narrowing the years to 30 or 33, provisionally judges 30 most likely.
[74] Newton to Richard Bentley 10 December 1692, in Turnbull et al. (1959-77), vol 3, p. 233.
[75] Opticks, 2nd Ed 1706. Query 31.

[76] H. G. Alexander (ed) The Leibniz-Clarke correspondence, Manchester University Press, 1998, p. 11.
[77] Jacob, Margaret C. (1976). The Newtonians and the English Revolution: 1689—1720. Cornell University Press, pp. 37, 44.

ISBN 0855270667.
[78] Westfall, Richard S. (1958). Science and Religion in Seventeenth-Century England. New Haven: Yale University Press, p. 200.

ISBN 0208008438.
[79] Haakonssen, Knud. "The Enlightenment, politics and providence: some Scottish and English comparisons". In Martin Fitzpatrick ed..

Enlightenment and Religion: Rational Dissent in eighteenth-century Britain. Cambridge: Cambridge University Press, p. 64.

ISBN 0521560608.
[80] Frankel, Charles (1948). The Faith of Reason: The Idea of Progress in the French Enlightenment. New York: King's Crown Press, p. 1.
[81] Germain, Gilbert G.. A Discourse on Disenchantment: Reflections on Politics and Technology, p. 28. ISBN 0791413195.
[82] Principia, Book III; cited in; Newton's Philosophy of Nature: Selections from his writings, p. 42, ed. H.S. Thayer, Hafner Library of

Classics, NY, 1953.
[83] A Short Scheme of the True Religion, manuscript quoted in Memoirs of the Life, Writings and Discoveries of Sir Isaac Newton by Sir David

Brewster, Edinburgh, 1850; cited in; ibid, p. 65.
[84] Webb, R.K. ed. Knud Haakonssen. "The emergence of Rational Dissent." Enlightenment and Religion: Rational Dissent in

eighteenth-century Britain. Cambridge University Press, Cambridge: 1996. pl9.
[85] H. G. Alexander (ed) The Leibniz-Clarke correspondence, Manchester University Press, 1998, p. 14.
[86] Westfall, 1958 p201.
[87] Marquard, Odo. "Burdened and Disemburdened Man and the Flight into Unindictability," in Farewell to Matters of Principle. Robert M.

Wallace trans. London: Oxford UP, 1989.
[88] Jacob, Margaret C. The Newtonians and the English Revolution: 1689-1720. plOO-101.
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christianpost.com/article/200706 19/28049_Papers_Show_Isaac_Newton's_Religious_Side,_Predict_Date_of_Apocalypse. htm). Associated

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28049_Papers_Show_Isaac_Newton's_Religious_Side,_Predict_Date_of_Apocalypse.htm) on 2007-08-13. . Retrieved 2007-08-01.
[90] Cassels, Alan. Ideology and International Relations in the Modern World. p2.
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of an ordered world clearly played a big part in the flowering of this movement in the eighteenth century" John Gribbin (2002) Science: A

History 1543-2001, p 241
[92] White 1997, p. 259
[93] White 1997, p. 267
[94] Westfall 2007, p.73
[95] White 1997, p 269
[96] Westfall 1994, p 229
[97] Westfall 1980, pp. 571-5
[98] Ball 1908, p. 337
[99] White 1997, p. 86
[100] Newton's apple: The real story (http://www.newscientist.com/blogs/culturelab/2010/01/newtons-apple-the-real-story.php). New

Scientist. 18 January 2010. . Retrieved 10 May 2010
[101] I. Bernard Cohen and George E. Smith, eds. The Cambridge Companion to Newton (2002) p. 6
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texts/normalized/THEM00167). Newtonproject. Imperial College London. . Retrieved 2006-08-30.
[103] Stukeley, William. "Memoirs of Sir Isaac Newton's Life" (http://physics.info/gravitation/apple.html). . Retrieved 2010-01-24.
[104] "Brogdale — Home of the National Fruit Collection" (http://www.brogdale.org/). Brogdale.org. . Retrieved 2008-12-20.

Isaac Newton 434

[105] "From the National Fruit Collection: Isaac Newton's Tree" (http://www.brogdale. org.uk/imagel. php?varietyid=1089). . Retrieved

2009-01-10.
[106] Newton's alchemical works (http://webappl.dlib.indiana.edu/newton/index.jsp) transcribed and online at Indiana University.

Retrieved 11 January 2007.
[107] http://www.pierre-marteau.com/editions/1701-25-mint-reports.html

References

• Ball, W.W. Rouse (1908). A Short Account of the History of Mathematics. New York: Dover. ISBN 0486206300.

• Christianson, Gale (1984). In the Presence of the Creator: Isaac Newton & His Times. New York: Free Press.
ISBN 0-02-905190-8. This well documented work provides, in particular, valuable information regarding
Newton's knowledge of Patristics

• Craig, John (1958). "Isaac Newton - Crime Investigator". Nature 182 (4629): 149-152. doi:10.1038/182149a0.

• Craig, John (1963). "Isaac Newton and the Counterfeiters". Notes and Records of the Royal Society of London 18
(2): 136-145. doi:10.1098/rsnr.l963.0017.

• Levenson, Thomas (2010). Newton and the Counterfeiter: The Unknown Detective Career of the World's Greatest
Scientist. Mariner Books. ISBN 9780547336046.

• Stewart, James (2009). Calculus: Concepts and Contexts. Cengage Learning. ISBN 9780495557425.

• Westfall, Richard S. (1980, 1998). Never at Rest. Cambridge University Press. ISBN 0-521-27435-4.

• Westfall, Richard S. (2007). Isaac Newton. Cambridge University Press. ISBN 9780199213559.

• Westfall, Richard S. (1994). The Life of Isaac Newton. Cambridge University Press. ISBN 0521477379.

• White, Michael (1997). Isaac Newton: The Last Sorcerer. Fourth Estate Limited. ISBN 1-85702-416-8.

Further reading

• Andrade, E. N. De C. (1950). Isaac Newton. New York: Chanticleer Press. ISBN 0841430144.

• Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time.
2006. 277 pp. excerpt and text search (http://www.amazon.com/dp/1560259922)

• Bechler, Zev (1991). Newton's Physics and the Conceptual Structure of the Scientific Revolution. Springer.
ISBN 0792310543..

• Berlinski, David. Newton's Gift: How Sir Isaac Newton Unlocked the System of the World. (2000). 256 pp.
excerpt and text search (http://www.amazon.com/dp/0743217764) ISBN 0-684-84392-7

• Buchwald, Jed Z. and Cohen, I. Bernard, eds. Isaac Newton's Natural Philosophy. MIT Press, 2001. 354 pp.
excerpt and text search (http://www.amazon.com/dp/0262524252)

• Casini, P. (1988). "Newton's Principia and the Philosophers of the Enlightenment" (http://links.jstor.org/
sici?sici=0035-9149(198801)42:l<35:N'ATPO>2.0.CO;2-H). Notes and Records of the Royal Society of London
42 (1): 35-52. doi: 10. 1098/rsnr. 1988.0006. ISSN 0035-9149.

• Christianson, Gale E. (1996). Isaac Newton and the Scientific Revolution. Oxford University Press.
ISBN 019530070X. See this site (http://www.amazon.com/dp/019530070X) for excerpt and text search.

• Christianson, Gale (1984). In the Presence of the Creator: Isaac Newton & His Times. New York: Free Press.
ISBN 0-02-905190-8.

• Cohen, I. Bernard and Smith, George E., ed. The Cambridge Companion to Newton. (2002). 500 pp. focuses on
philosophical issues only; excerpt and text search (http://www.amazon.com/dp/0521656966); complete
edition online (http://www.questia.com/read/105054986)

• Cohen, I. B. (1980). The Newtonian Revolution. Cambridge: Cambridge University Press. ISBN 0521229642.

• Craig, John (1946). Newton at the Mint. Cambridge, England: Cambridge University Press.

• Dampier, William C; Dampier, M. (1959). Readings in the Literature of Science. New York: Harper & Row.
ISBN 0486428052.

Isaac Newton 435

de Villamil, Richard (1931). Newton, the Man. London: G.D. Knox. — Preface by Albert Einstein. Reprinted by

Johnson Reprint Corporation, New York (1972).

Dobbs, B. J. T. (1975). The Foundations of Newton's Alchemy or "The Hunting of the Greene Lyon". Cambridge:

Cambridge University Press.

Gjertsen, Derek (1986). The Newton Handbook. London: Routledge & Kegan Paul. ISBN 0-7102-0279-2.

Gleick, James (2003). Isaac Newton. Alfred A. Knopf. ISBN 0375422331.

Halley, E. (1687). "Review of Newton's Principia". Philosophical Transactions 186: 291—297.

Hawking, Stephen, ed. On the Shoulders of Giants. ISBN 0-7624-1348-4 Places selections from Newton's

Principia in the context of selected writings by Copernicus, Kepler, Galileo and Einstein

Herivel, J. W. (1965). The Background to Newton's Principia. A Study of Newton's Dynamical Researches in the

Years 1664—84. Oxford: Clarendon Press.

Keynes, John Maynard (1963). Essays in Biography. W. W. Norton & Co. ISBN 0-393-00189-X. Keynes took a

close interest in Newton and owned many of Newton's private papers.

Koyre, A. (1965). Newtonian Studies. Chicago: University of Chicago Press.

Newton, Isaac. Papers and Letters in Natural Philosophy, edited by I. Bernard Cohen. Harvard University Press,

1958,1978. ISBN 0-674-46853-8.

Newton, Isaac (1642-1727). The Principia: a new Translation, Guide by I. Bernard Cohen ISBN 0-520-08817-4

University of California (1999)

Pemberton, H. (1728). A View of Sir Isaac Newton's Philosophy . London: S. Palmer.

Shamos, Morris H. (1959). Great Experiments in Physics. New York: Henry Holt and Company, Inc..

ISBN 0486253465.

Shapley, Harlow, S. Rapport, and H. Wright. A Treasury of Science; "Newtonia" pp. 147—9; "Discoveries"

pp. 150-4. Harper & Bros., New York, (1946).

Simmons, J. (1996). The Giant Book of Scientists — The 100 Greatest Minds of all Time. Sydney: The Book

Company.

Stukeley, W. (1936). Memoirs of Sir Isaac Newton's Life. London: Taylor and Francis, (edited by A. H. White;

originally published in 1752)

Westfall, R. S. (1971). Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century. London:

Macdonald. ISBN 0444196110.

Religion

• Dobbs, Betty Jo Tetter. The Janus Faces of Genius: The Role of Alchemy in Newton's Thought. (1991), links the
alchemy to Arianism

• Force, James E., and Richard H. Popkin, eds. Newton and Religion: Context, Nature, and Influence. (1999),
342pp . Pp. xvii + 325. 13 papers by scholars using newly opened manuscripts

• Ramati, Ayval. "The Hidden Truth of Creation: Newton's Method of Fluxions" British Journal for the History of
Science 34: 417—438. in JSTOR (http://www.jstor.org/stable/4028372), argues that his calculus had a
theological basis

• Snobelen, Stephen '"God of Gods, and Lord of Lords': The Theology of Isaac Newton's General Scholium to the
Principia," Osiris, 2nd Series, Vol. 16, (2001), pp. 169-208 in JSTOR (http://www.jstor.org/stable/301985)

• Snobelen, Stephen D. "Isaac Newton, Heretic: The Strategies of a Nicodemite," British Journal for the History of
Science 32: 381-419. in JSTOR (http://www.jstor.org/stable/4027945)

• Pfizenmaier, Thomas C. "Was Isaac Newton an Arian?," Journal of the History of Ideas, Vol. 58, No. 1 (January,
1997), pp. 57-80 in JSTOR (http://www.jstor.org/stable/3653988)

• Wiles, Maurice. Archetypal Heresy. Arianism through the Centuries. (1996) 214pp, with chapter 4 on 18th
century England; pp 77—93 on Newton excerpt and text search (http://books.google.com/
books?id=DGksMzk37hMC&printsec=frontcover&dq="Arianism+through+the+Centuries"),

Isaac Newton 436

Primary sources

• Newton, Isaac. The Principia: Mathematical Principles of Natural Philosophy. University of California Press,
(1999). 974 pp.

• Brackenridge, J. Bruce. The Key to Newton's Dynamics: The Kepler Problem and the Principia: Containing an
English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton's
Mathematical Principles of Natural Philosophy. University of California Press, 1996. 299 pp.

• Newton, Isaac. The Optical Papers of Isaac Newton. Vol. 1: The Optical Lectures, 1670—1672. Cambridge U.
Press, 1984. 627 pp.

• Newton, Isaac. Opticks (4th ed. 1730) online edition (http://books.google.com/
books?id=GnAFAAAAQAAJ&dq=newton+opticks&pg=PPl&ots=Nnl345oqo_&
sig=0mBTaXUI_K6w-JDEu_RvVq5TNqc&prev=http://www. google. com/search?q=newton+opticks&
rls=com.microsoft:en-us:IE-SearchBox&ie=UTF-8&oe=UTF-8&sourceid=ie7&rlz=H7GGLJ&sa=X&
oi=print& ct=title& cad=one-book- with-thumbnail)

• Newton, I. (1952). Opticks, or A Treatise of the Reflections, Refractions, Inflections & Colours of Light. New
York: Dover Publications.

• Newton, I. Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World, tr. A.
Motte, rev. Florian Cajori. Berkeley: University of California Press. (1934).

• Whiteside, D. T. (1967—82). The Mathematical Papers of Isaac Newton. Cambridge: Cambridge University Press.
ISBN 0521077400. - 8 volumes

• Newton, Isaac. The correspondence of Isaac Newton, ed. H. W. Turnbull and others, 7 vols. (1959—77)

• Newton's Philosophy of Nature: Selections from His Writings edited by H. S. Thayer, (1953), online edition (http:/
/www. questia.com/read/5876270)

• Isaac Newton, Sir; J Edleston; Roger Cotes, Correspondence of Sir Isaac Newton and Professor Cotes, including
letters of other eminent men (http://books. google. com/books?as_brr=l&id=OVPJ6c9_kKgC&

vid=OCLC 1443778 l&dq="isaac+newton"&jtp=I), London, John W. Parker, West Strand; Cambridge, John
Deighton, 1850. - Google Books

• Maclaurin, C. (1748). An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books. London: A.
Millar and J. Nourse.

• Newton, I. (1958). Isaac Newton's Papers and Letters on Natural Philosophy and Related Documents, eds. I. B.
Cohen and R. E. Schofield. Cambridge: Harvard University Press.

• Newton, I. (1962). The Unpublished Scientific Papers of Isaac Newton: A Selection from the Portsmouth
Collection in the University Library, Cambridge, ed. A. R. Hall and M. B. Hall. Cambridge: Cambridge
University Press.

• Newton, I. (1975). Isaac Newton's 'Theory of the Moon's Motion' (1702). London: Dawson.

External links

• @ Chisholm, Hugh, ed (1911). "Newton, Sir Isaac". Encyclopedia Britannica (Eleventh ed.). Cambridge
University Press.

• ScienceWorld biography (http://scienceworld.wolfram.com/biography/Newton.html) by Eric Weisstein

• Dictionary of Scientific Biography (http://www.chlt.Org/sandbox/lhl/dsb/page.50.a.php)

• The Newton Project (http://www.newtonproject.sussex.ac.uk/prism.php?id=l)

• The Newton Project - Canada (http://www.isaacnewton.ca/)

• Rebuttal of Newton's astrology (http://web.archive.Org/web/20080629021908/http://www.skepticreport.
com/predictions/newton.htm) (via archive.org)

• Newton's Religious Views Reconsidered (http://www.galilean-library.org/snobelen.html)

• Newton's Royal Mint Reports (http://www.pierre-marteau.com/editions/1701-25-mint-reports.html)

Isaac Newton 437

• Newton's Dark Secrets (http://www.pbs.org/wgbh/nova/newton/) NOVA TV programme

• from The Stanford Encyclopedia of Philosophy:

• Isaac Newton (http://plato.stanford.edu/entries/newton/), by George Smith

• Newton's Philosophiae Naturalis Principia Mathematica (http://plato.stanford.edu/entries/newton-principia/
), by George Smith

• Newton's Philosophy (http://plato.stanford.edu/entries/newton-philosophy/), by Andrew Janiak

• Newton's views on space, time, and motion (http://plato.stanford.edu/entries/newton-stm/), by Robert
Rynasiewicz

Newton's Castle (http://www.tqnyc.org/NYC051308/index.htm) Educational material

The Chymistry of Isaac Newton (http://www.dlib.indiana.edu/collections/newton) Research on his

Alchemical writings

FMA Live! Program for teaching Newton's laws to kids (http://www.fmalive.com/)

Newton's religious position (http://www.adherents.com/people/pn/Isaac_Newton.html)

The "General Scholium" to Newton's Principia (http://hss.fullerton.edu/philosophy/GeneralScholium.htm)

Kandaswamy, Anand M. The Newton/Leibniz Conflict in Context (http://www.math.rutgers.edu/courses/436/

Honors02/newton. html)

Newton's First ODE (http://www.phaser.com/modules/historic/newton/index.html) — A study by on how

Newton approximated the solutions of a first-order ODE using infinite series

O'Connor, John J.; Robertson, Edmund F., "Isaac Newton" (http://www-history.mcs.st-andrews.ac.uk/

Biographies/Newton, html), MacTutor History of Mathematics archive, University of St Andrews.

Isaac Newton (http://www. genealogy. ams.org/id.php?id=74313) at the Mathematics Genealogy Project

The Mind of Isaac Newton (http://www.ltrc.mcmaster.ca/newton/) Images, audio, animations and interactive

segments

Enlightening Science (http://www.enlighteningscience.sussex.ac.uk/home) Videos on Newton's biography,

optics, physics, reception, and on his views on science and religion

Newton biography (University of St Andrews) (http://www-history.mcs.st-andrews.ac.uk/Mathematicians/

Newton.html)

Writings by him

• Newton's works - full texts, at the Newton Project (http://www.newtonproject.sussex.ac.uk/prism.php?id=43)

• Works by Isaac Newton (http://www.gutenberg.org/author/Isaac_Newton) at Project Gutenberg

• Newton's Principia — read and search (http://rackl.ul.cs.cmu.edu/is/newton/)

• Descartes, Space, and Body and A New Theory of Light and Colour (http://www.earlymoderntexts.com/),
modernised readable versions by Jonathan Bennett

• Opticks, or a Treatise of the Reflections, Refractions, Inflexions and Colours of Light (http://www.archive.org/
stream/opticksoratreat00newtgoog#page/n6/mode/2up), full text on archive.org

Bernhard Riemann

438

Bernhard Riemann

Bernhard Riemann, 1 863

Born

September 17, 1826Breselenz, Kingdom of Hanover (modern-day Germany)

Died

July 20, 1866 (aged 39)Selasca, Kingdom of Italy

Residence

Kingdom of Hanover

Nationality

German

Fields

Mathematics

Institutions

Georg- August University of Gottingen

Alma mater

Georg- August University of Gottingen
Berlin University

Doctoral advisor

Carl Friedrich Gauss

Other academic advisors

Gotthold Eisenstein
Moritz Abraham Stern

Notable students

Gustav Roch

Known for

See list

Influences

Johann Peter Gustav Lejeune Dirichlet

Georg Friedrich Bernhard Riemann (German pronunciation: ['Ki'.man]; September 17, 1826 — July 20, 1866) was an
influential German mathematician who made lasting contributions to analysis and differential geometry, some of
them enabling the later development of general relativity.

Biography

Early years

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal
Republic of Germany today. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who
fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood.
Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns. Riemann exhibited
exceptional mathematical skills, such as fantastic calculation abilities, from an early age but suffered from timidity
and a fear of speaking in public.

Bernhard Riemann 439

Education

During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school). After the
death of his grandmother in 1842, he attended high school at the Johanneum Luneburg. In high school, Riemann
studied the Bible intensively, but he was often distracted by mathematics. To this end, he even tried to prove
mathematically the correctness of the Book of Genesis. His teachers were amazed by his adept ability to solve
complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of
19, he started studying philology and theology in order to become a priest and help with his family's finances.

During the spring of 1846, his father (Friedrich Riemann), after gathering enough money to send Riemann to
university, allowed him to stop studying theology and start studying mathematics. He was sent to the renowned
University of Gottingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least
squares.

In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, Steiner, and Eisenstein were teaching. He stayed in
Berlin for two years and returned to Gottingen in 1849.

Academia

Bernhard Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set
the stage for Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to
extraordinary professor status at the University of Gottingen. Although this attempt failed, it did result in Riemann
finally being granted a regular salary. In 1859, following Dirichlet's death, he was promoted to head the mathematics
department at Gottingen. He was also the first to suggest using dimensions higher than merely three or four in order
to describe physical reality — an idea that was ultimately vindicated with Einstein's contribution in the early 20th
century. In 1862 he married Elise Koch and had a daughter.

Austro-Prussian War

Riemann fled Gottingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis
during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the
cemetery in Biganzolo (Verbania). Meanwhile, in Gottingen his housekeeper tidied up some of the mess in his
office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may
have been lost forever.

Influence

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently
become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The
theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics
is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.

Riemann made major contributions to real analysis. He defined the Riemann integral by means of Riemann sums,
developed a theory of trigonometric series that are not Fourier series — a first step in generalized function
theory — and studied the Riemann— Liouville differintegral.

He made some famous contributions to modern analytic number theory. In a single short paper (the only one he
published on the subject of number theory), he introduced the Riemann zeta function and established its importance
for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta
function, one of which is the well-known Riemann hypothesis.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful
heuristic rather than a rigorous method. Its justification took at least a generation. His work on monodromy and the
hypergeometric function in the complex domain made a great impression, and established a basic way of working

Bernhard Riemann 440

with functions by consideration only of their singularities.

Euclidean geometry versus Riemannian geometry

In 1853, Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over
many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture at
Gottingen in 1854, the mathematical public received it with enthusiasm, and it is one of the most important works in
geometry. It was titled Uber die Hypothesen welche der Geometrie zu Grunde liegen ("On the hypotheses which
underlie geometry"), and was published in 1868.

The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n
dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The
fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number
(scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean
geometries.

Higher dimensions

Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe
how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten
numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous
construction central to his geometry, known now as a Riemannian metric.

Writings in English

• 1868 "On the hypotheses which lie at the foundation of geometry" translated by W.K.Clifford, Nature 8 1873 183-
reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea).

• 1868. "On the hypotheses which lie at the foundation of geometry" in Ewald, William B., ed., 1996. "From Kant to
Hilbert: A Source Book in the Foundations of Mathematics" , 2 vols. Oxford Uni. Press: 652—61.

• Riemann, Bernhard (2004), Collected papers, Kendrick Press, Heber City, UT, ISBN 978-0-9740427-2-5,
MR2121437

Notes

[1] Werke, p. 268, edition of 1876, cited in Pierpont, Non-Euclidean Geometry, A Retrospect (http://projecteuclid.org/euclid.bams/

1183493815)
[2] Marcus du Sautoy, The Music of the Primes, (HarperCollins 2003)

Further reading

• Derbyshire, John (2003), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in
Mathematics, Washington, DC: John Henry Press, ISBN 0309085497.

• Monastyrsky, Michael (1999), Riemann, Topology and Physics, Boston, MA: Birkhauser, ISBN 0817637893.

External links

• Bernhard Riemann (http://www. genealogy. ams.org/id.php?id=18232) at the Mathematics Genealogy Project

• The Mathematical Papers of Georg Friedrich Bernhard Riemann (http://www.maths.tcd.ie/pub/HistMath/
People/Riemann/Papers . html)

• All publications of Riemann can be found at: http://www.emis.de/classics/Riemann/

Bernhard Riemann 441

• O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Riemann.html), MacTutor History of Mathematics archive, University of St Andrews.

• Bernhard Riemann — one of the most important mathematicians (http://www.fh-lueneburg.de/ul/gym03/
englpage/chronik/riemann/riemann.htm)

• Bernhard Riemann's inaugural lecture (http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/
WKCGeom.html)

• Weisstein, Eric W., Riemann, Bernhard (1826—1866) (http://scienceworld.wolfram.com/biography/Riemann.
html) from ScienceWorld.

Jean Dieudonne

442

Jean Dieudonne

Jean Alexandre Eugene Dieudonne

Born

Jean Alexandre Eugene Dieudonne
1 July 1906Lille, France

Died
Nationality

Fields
Institutions

Alma mater
Doctoral advisor

Doctoral students

Known for

29 November 1992 (aged 86)Paris, France
| France

Mathematics

University of Sao Paulo

University of Nancy

University of Michigan

Northwestern University

Institut des Hautes Etudes Scientifiques

University of Nice

Ecole Normale Superieure
Paul Montel

Edmond Fedida
Alexander Grothendieck
Kishore Marathe

Cartan— Dieudonne theorem

Jean Alexandre Eugene Dieudonne (1 July 1906 — 29 November 1992) was a French mathematician, notable for
research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous
group and the Elements de ge'ome'trie alge'brique project of Alexander Grothendieck, and as a historian of
mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups
(the book La Geometrie des groupes classiques was published in 1955), and on formal groups, introducing what now
are called Dieudonne modules, had a major effect on those fields.

He was born and brought up in Lille, with a formative stay in England where he was introduced to algebra. In 1924
he was accepted for the Ecole Normale Superieure, where Andre Weil was a contemporary. He began working,
conventionally enough, in complex analysis. In 1934 he was one of the group of normaliens convened by Weil,
which would become 'Bourbaki'.

Jean Dieudonne 443

Education and teaching

He served in the French Army in World War II, and then taught in Clermont-Ferrand until the liberation of France.
After holding professorships at the University of Sao Paulo (1946—47), the University of Nancy (1948—1952) and the
University of Michigan (1952—53), he joined the Department of Mathematics at Northwestern University in 1953,
before returning to France as a founding member of the Institut des Hautes Etudes Scientifiques. He moved to the
University of Nice to found the Department of Mathematics in 1964, and retired in 1970. He was elected as a
member of the Academie des Sciences in 1968.

Career

He drafted much of the Bourbaki series of texts, the many volumes of the EGA algebraic geometry series, and nine
volumes of his own Traite d 'Analyse. The first volume of the Traite is a French translation of the book Foundations
of Modern Analysis (1960), which had become a graduate textbook on functional analysis.

He also wrote individual monographs on Infinitesimal Calculus, Linear Algebra and Elementary Geometry, invariant
theory, commutative algebra, algebraic geometry, and formal groups.

With Laurent Schwartz he supervised the early research of Alexander Grothendieck; later from 1959 to 1964 he was
at IHES alongside Grothendieck, and collaborating on the expository work needed to support the project of
refounding algebraic geometry on the new basis of schemes.

Works

• Dieudonne, Jean (1955), La geometrie des groupes classiques , Ergebnisse der Mathematik und ihrer
Grenzgebiete (N.F.), Heft 5, Berlin, New York: Springer- Verlag, ISBN 978-0-387-05391-2, MR0072144

• Foundations of Modern Analysis (1960), Academic Press

• Dieudonne, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,
doi: 10. 1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR0279102 (a reprint of Dieudonne, Jean A.;
Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics 4: 1—80,

doi: 10. 1016/0001-8708(70)90015-0, ISSN 0001-8708, MR0255525)

• Dieudonne, Jean Alexandre (1982), A panorama of pure mathematics , Pure and Applied Mathematics, 97,
London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-215560-4, MR0478177

• Dieudonne, Jean (1981) (in French), Choix d'ceuvres mathematiques. Tome I, Paris: Hermann,
ISBN 978-2-7056-5922-6, MR61 1 149

• Dieudonne, Jean (1981) (in French), Choix d'ceuvres mathematiques. Tome II, Paris: Hermann,
ISBN 978-2-7056-5923-3, MR61 1 150

References

• Dugac, Pierre (1995) (in French), Jean Dieudonne: Mathematicien complet (Plus de lumiere) , Editions Jacques
Gabay, ISBN 978-2876471566

External links

• O'Connor, John J.; Robertson, Edmund F., "Jean Dieudonne" , MacTutor History of Mathematics archive,

University of St Andrews.

• Jean Dieudonne at the Mathematics Genealogy Project

Jean Dieudonne

444

References

[1] http:

[2] http:

[3] http

[4] http:

[5] http:

//books. google.com/books?id=AfYZAQAAIAAJ
//books.google.com/books?isbn=978-0-12-215560-4
//www.gabay.com/sources/Liste_Fiche.asp?CV=76
//www-history. mcs.st-andrews.ac.uk/Biographies/Dieudonne. html
//www. genealogy. ams.org/id.php?id=342 19

Alexander Grothendieck

445

Alexander Grothendieck

Alexander Grothendieck in Montreal, 1970

Born

28 March 1928Berlin, Germany

Residence

France

Nationality

None (Stateless)

Fields

Mathematics

Institutions

Institut des Hautes Etudes Scientifiques

Alma mater

University of Montpellier

University of Nancy

Doctoral advisor

Laurent Schwartz

Doctoral students

Pierre Berthelot

Pierre Deligne

Michel Demazure

Jean Giraud

Luc Illusie

Michel Raynaud

Jean-Louis Verdier

Notable awards

Fields Medal (1966)

Crafoord Prize (1988, declined)

Alexander Grothendieck (born 28 March 1928) is a mathematician and the central figure behind the creation of the
modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating
major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations.
This new perspective led to revolutionary advances across many areas of pure mathematics.

Within algebraic geometry itself, his theory of schemes has become the universally accepted language for all further
technical work. His generalization of the classical Riemann-Roch theorem launched the study of algebraic and
topological K-theory. His construction of new cohomology theories has left deep consequences for algebraic number
theory, algebraic topology, and representation theory. His creation of topos theory has had an impact on set theory
and logic.

One of his most celebrated achievements is the discovery of the first arithmetic Weil cohomology theory: the ^-adic
etale cohomology. This key result opened the way for a proof of the Weil conjectures, ultimately completed by his
student Pierre Deligne. To this day, ^-adic cohomology remains a fundamental tool for number theorists, with
important applications to the Langlands program.

Grothendieck' s way of thinking has influenced generations of mathematicians long after his departure from
mathematics. His emphasis on the role of universal properties brought category theory into the mainstream as an

Alexander Grothendieck 446

important organizing principle. His notion of abelian category is now the basic object of study in homological
algebra. His conjectural theory of motives has been a driving force behind modern developments in algebraic
K-theory, motivic homotopy theory, and motivic integration.

Driven by deep personal and political convictions, Grothendieck left the Institut des Hautes Etudes Scientifiques,
where he had been appointed professor and accomplished his greatest work, after a dispute over military funding in
1970. His mathematical activity essentially ceased after this, and he devoted his energies to political causes. He
formally retired in 1988 and within a few years moved to the Pyrenees, where he currently lives in isolation from
human society.

Mathematical achievements

Grothendieck's early mathematical work was in functional analysis. Between 1949 and 1953 he worked on his
doctoral thesis in this subject at Nancy, supervised by Jean Dieudonne and Laurent Schwartz. His key contributions
include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for
Schwartz distributions, and the application of L p spaces in studying linear maps between topological vector spaces.
In a few years, he had turned himself into a leading authority on this area of functional analysis — to the extent that
Dieudonne compares his impact in this field to that of Banach.

It is, however, in algebraic geometry and related fields where Grothendieck did his most important and influential
work. From about 1955 he started to work on sheaf theory and homological algebra, producing the influential
"Tohoku paper" {Sur quelques points d'algebre homologique, published in 1957) where he introduced Abelian
categories and applied their theory to show that sheaf cohomology can be defined as certain derived functors in this
context.

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and
others, after sheaves had been defined by Jean Leray. Grothendieck took them to a higher level of abstraction and
turned them into a key organising principle of his theory. He shifted attention from the study of individual varieties
to the relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many
classical theorems. The first major application was the relative version of Serre's theorem showing that the
cohomology of a coherent sheaf on a complete variety is finite dimensional; Grothendieck's theorem shows that the
higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a
one-point space.

In 1956, he applied the same thinking to the Riemann— Roch theorem, which had already recently been generalized to
any dimension by Hirzebruch. The Grothendieck— Riemann— Roch theorem was announced by Grothendieck at the
initial Mathematische Arbeitstagung in Bonn, in 1957. It appeared in print in a paper written by Armand Borel with
Serre. This result was his first major achievement in algebraic geometry. He went on to plan and execute a major
foundational programme for rebuilding the foundations of algebraic geometry, which were then in a state of flux and
under discussion in Claude Chevalley's seminar; he outlined his programme in his talk at the 1958 International
Congress of Mathematicians.

His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted
the use of non-closed generic points, which led to the theory of schemes. He also pioneered the systematic use of
nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic
settings. His theory of schemes has become established as the best universal foundation for this major field, because
of its great expressive power as well as technical depth. In that setting one can use birational geometry, techniques
from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic
topology, all in an integrated way.

He is also noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of
formulation and presentation. Relatively little of his work after 1960 was published by the conventional route of the
learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable

Alexander Grothendieck 447

extent personal. His influence spilled over into many other branches of mathematics, for example the contemporary
theory of D-modules. (It also provoked adverse reactions, with many mathematicians seeking out more concrete
areas and problems.)

EGA and SGA

The bulk of Grothendieck's published work is collected in the monumental, and yet incomplete, Elements de
geometrie algebrique (EGA) and Seminaire de geometrie algebrique (SGA). The collection Fondements de la
Geometrie Algebrique (FGA), which gathers together talks given in the Seminaire Bourbaki, also contains important
material.

Perhaps Grothendieck's deepest single accomplishment is the invention of the etale and 1-adic cohomology theories,
which explain an observation of Andre Weil's that there is a deep connection between the topological characteristics
of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation
over a finite field reflects the topological nature of its solutions over the complex numbers. Weil realized that to
prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do
this until such a theory was found by Grothendieck.

This program culminated in the proofs of the Weil conjectures, the last of which was settled by Grothendieck's
student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.

Major mathematical topics (from Recoltes et Semailles)

He wrote a retrospective assessment of his mathematical work (see the external link La Vision below). As his main
mathematical achievements ("maitre-themes"), he chose this collection of 12 topics (his chronological order):

1 . Topological tensor products and nuclear spaces

2. "Continuous" and "discrete" duality (derived categories and "six operations").

3. Yoga of the Grothendieck— Riemann— Roch theorem (K-theory, relation with intersection theory).

4. Schemes.

5. Topoi.

6. Etale cohomology including 1-adic cohomology.

7. Motives and the motivic Galois group (and Grothendieck categories)

8. Crystals and crystalline cohomology, yoga of De Rham and Hodge coefficients.

9. Topological algebra, infinity-stacks, 'derivateurs', cohomological formalism of toposes as an inspiration for a new
homotopic algebra

10. Tame topology.

11. Yoga of anabelian geometry and Galois— Teichmuller theory.

12. Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.

He wrote that the central theme of the topics above is that of topos theory, while the first and last were of the least
importance to him.

Here the term yoga denotes a kind of "meta-theory" that can be used heuristically; Michel Raynaud writes the other
terms "Ariadne's thread" and "philosophy" as effective equivalents.

Alexander Grothendieck 448

Life

Family and early life

Alexander Grothendieck was born in Berlin to anarchist parents: a Ukrainian father from an ultimately Hassidic
family, Alexander "Sascha" Shapiro aka Tanaroff, and a mother from a German Protestant family, Johanna "Hanka"

ro]

Grothendieck; both of his parents had broken away from their early backgrounds in their teens. At the time of his
birth Grothendieck's mother was married to Johannes Raddatz, a German journalist, and his birthname was initially
recorded as Alexander Raddatz. The marriage was dissolved in 1929 and Shapiro/Tanaroff acknowledged his
paternity, but never married Hanka Grothendieck. Grothendieck lived with his parents until 1933 in Berlin. At the

end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Grothendieck in the care
of Wilhelm Heydorn, a Lutheran Pastoi
parents fought in the Spanish Civil War.

of Wilhelm Heydorn, a Lutheran Pastor and teacher in Hamburg where he went to school. During this time, his

During WWII

In 1939 Grothendieck came to France and lived in various camps for displaced persons with his mother, first at the
Camp de Rieucros, and subsequently lived for the remainder of the war in the village of Le Chambon-sur-Lignon,
where he was sheltered and hidden in local boarding-houses or pensions. His father was sent via Drancy to
Auschwitz where he died in 1942. While Grothendieck lived in Chambon, he attended the College Cevenol (now
known as the Le College-Lycee Cevenol International), a unique secondary school founded in 1938 by local
Protestant pacifists and anti-war activists. Many of the refugee children being hidden in Chambon attended Cevenol
and it was at this school that Grothendieck apparently first became fascinated with mathematics.

Studies and contact with research mathematics

After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier.
After three years of increasingly independent studies there he got a scholarship to go to continue his studies in Paris
inl948 [10]

Initially, Grothendieck attended Henri Cartan's Seminar at Ecole Normale Superieure, but lacked the necessary
background to follow the high-powered seminar. On the advice of Cartan and Weil, he moved to the University of
Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950 to 1953. At this
time he was a leading expert in the theory of topological vector spaces. By 1957, he set this subject aside in order to
work in algebraic geometry and homological algebra.

The IHES years

Installed at the Institut des Hautes Etudes Scientifiques (IHES), Grothendieck attracted attention by an intense and
highly productive activity of seminars {de facto working groups drafting into foundational work some of the ablest
French and other mathematicians of the younger generation). Grothendieck himself practically ceased publication of
papers through the conventional, learned journal route. He was, however, able to play a dominant role in
mathematics for around a decade, gathering a strong school.

During this time he had officially as students Michel Demazure (who worked on SGA3, on group schemes), Luc
Illusie (cotangent complex), Michel Raynaud, Jean-Louis Verdier (cofounder of the derived category theory) and
Pierre Deligne. Collaborators on the SGA projects also included Mike Artin (etale cohomology) and Nick Katz
(monodromy theory and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of non-abelian
cohomology. Many others were involved.

Alexander Grothendieck 449

The 'Golden Age'

Alexander Grothendieck's work during the "Golden Age' period at IHES established several unifying themes in
algebraic geometry, number theory, topology, category theory and complex analysis. His first (pre-IHES)
breakthrough in algebraic geometry was the Grothendieck— Hirzebruch— Riemann— Roch theorem, a far-reaching
generalisation of the Hirzebruch— Riemann— Roch theorem proved algebraically; in this context he also introduced
K-theory. Then, following the programme he outlined in his talk at the 1958 International Congress of
Mathematicians, he introduced the theory of schemes, developing it in detail in his Elements de geometrie algebrique
(EGA) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in
the field since that time. He went on to introduce the etale cohomology theory of schemes, providing the key tools
for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to
complement it. Closely linked to these cohomology theories, he originated topos theory as a generalisation of
topology (relevant also in categorical logic). He also provided an algebraic definition of fundamental groups of
schemes and more generally the main structures of a categorical Galois theory. As a framework for his coherent
duality theory he also introduced derived categories, which were further developed by Verdier.

The results of work on these and other topics were published in the EGA and in less polished form in the notes of the
Seminaire de geometrie algebrique (SGA) that he directed at IHES.

Politics and retreat from scientific community

Grothendieck's political views were radical and pacifist. Thus he strongly opposed both United States aggression in
Vietnam and Soviet military expansionism. He gave lectures on category theory in the forests surrounding Hanoi
while the city was being bombed, to protest against the Vietnam War (The Life and Work of Alexander
Grothendieck, American Mathematical Monthly, vol. 113, no. 9, footnote 6). He retired from scientific life around
1970, after having discovered the partly military funding of IHES (see pp. xii and xiii of SGA1, Springer Lecture
Notes 224). He returned to academia a few years later as a professor at the University of Montpellier, where he
stayed until his retirement in 1988. His criticisms of the scientific community, and especially of several mathematics
circles, are also contained in a letter, written in 1988, in which he states the reasons for his refusal of the Crafoord
Prize. He declined the prize on ethical grounds in an open letter to the media.

While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from
IHES, those who knew him say that the causes of the rupture ran deeper. Pierre Cartier, a visiteur de longue duree
("long-term guest") at the IHES, wrote a piece about Grothendieck for a special volume published on the occasion of
the IHES's fortieth anniversary. The Grothendieck Festschrift was a three-volume collection of research papers to

ri3i

mark his sixtieth birthday (falling in 1988), and published in 1990.

In it Cartier notes that, as the son of an antimilitary anarchist and one who grew up among the disenfranchised,
Grothendieck always had a deep compassion for the poor and the downtrodden. As Cartier puts it, Grothendieck
came to find Bures-sur-Yvette "une cage doree" ("a golden cage"). While Grothendieck was at the IHES, opposition
to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having
become a mandarin of the scientific world. In addition, after several years at the IHES Grothendieck seemed to cast
about for new intellectual interests. By the late 1960s he had started to become interested in scientific areas outside
of mathematics. David Ruelle, a physicist who joined the IHES faculty in 1964, said that Grothendieck came to talk
to him a few times about physics. (In the 1970s Ruelle and the Dutch mathematician Floris Takens produced a new
model for turbulence, and it was Ruelle who invented the concept of a strange attractor in a dynamical system.)

ri4i

Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics.

After leaving the IHES, Grothendieck became a temporary professor at College de France for two years. A
permanent position became open at the end of his tenure, but the application Grothendieck submitted made it clear
that he had no plans to continue his mathematical research. The position was given to Jacques Tits.

Alexander Grothendieck 450

He then went to Universite de Montpellier, where he became increasingly estranged from the mathematical
community. Around this time, he founded a group called Survivre, which was dedicated to antimilitary and
ecological issues. His mathematical career, for the most part, ended when he left the IHES. In 1984 he wrote a
proposal to get a position through the Centre National de la Recherche Scientifique. The proposal, entitled Esquisse
d'un Programme ("Program Sketch") describes new ideas for studying the moduli space of complex curves.
Although Grothendieck himself never published his work in this area, the proposal became the inspiration for work
by other mathematicians and the source of the theory of dessin d'enfants. Esquisse d'un Programme was published in
the two-volume proceedings Geometric Galois Actions (Cambridge University Press, 1997).'

Manuscripts written in the 1980s

While not publishing mathematical research in conventional ways during the 1980s, he produced several influential
manuscripts with limited distribution, with both mathematical and biographical content. During that period he also
released his work on Bertini type theorems contained in EGA 5, published by the Grothendieck Circle in 2004.

La Longue Marche a travers la theorie de Galois [The Long March Through Galois Theory] is an approximately
1600-page handwritten manuscript produced by Grothendieck during the years 1980—1981, containing many of the

ri7i

ideas leading to the Esquisse d'un programme (see below, and also a more detailed entry), and in particular
studying the Teichmuller theory.

In 1983 he wrote a huge extended manuscript (about 600 pages) entitled Pursuing Stacks, stimulated by

no]

correspondence with Ronald Brown, (see also R.Brown and Tim Porter at University of Bangor in Wales), and
starting with a letter addressed to Daniel Quillen. This letter and successive parts were distributed from Bangor (see
External Links below): in an informal manner, as a kind of diary, Grothendieck explained and developed his ideas on
the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative
theory of stacks. The manuscript, which is being edited for publication by G Maltsiniotis, later led to another of his
monumental works, Les Derivateurs. Written in 1991, this latter opus of about 2000 pages further developed the
homotopical ideas begun in Pursuing Stacks. Much of this work anticipated the subsequent development of the
motivic homotopy theory of Fabien Morel and V. Voevodsky in the mid 1990s.

ri7i

His Esquisse d'un programme (1984) is a proposal for a position at the Centre National de la Recherche
Scientifique, which he held from 1984 to his retirement in 1988. Ideas from it have proved influential, and have been
developed by others, in particular dessins d'enfants and a new field emerging as anabelian geometry. In La Clef des
Songes he explains how the reality of dreams convinced him of God's existence.

The 1000-page autobiographical manuscript Recoltes et semailles (1986) is now available on the internet in the
French original, and an English translation is underway (these parts of Recoltes et semailles have already been

translated into Russian] and published in Moscow ). Some parts of Recoltes et semailles and the whole La

T221
Clef des Songes have been translated into Spanish.

Retirement into reclusion

Grothendieck was co-awarded (but declined) the Crafoord Prize with Pierre Deligne in 1988.

In 1991, Grothendieck moved to an address he did not provide to his previous contacts in the mathematical
community. He is now said to live in southern France or Andorra and to be reclusive.

In January 2010, Grothendieck wrote a letter to Luc Illusie. In this "Declaration d'intention de non-publication", he
states that essentially all materials that have been published in his absence have been done without his permission.

He asks that none of his work should be reproduced in whole or in part, and even further that libraries containing

T231
such copies of his work remove them.

Alexander Grothendieck 45 1

Notes

[I] (Dieudonne 1990)

[2] See, for example, (Deligne 1998).

[3] Jackson, Allyn (2004), "Comme Appele du Neant — As If Summoned from the Void: The Life of Alexandre Grothendieck I" (http://www.

ams.org/notices/200409/fea-grothendieck-partl.pdf) (PDF), Notices of the American Mathematical Society 51 (4): 1049,
[4] Mclarty, Colin. "The Rising Sea: Grothendieck on simplicity and generality I" (http://people.math.jussieu.fr/~leila/grothendieckcircle/

mclartyl.pdf) (PDF). . Retrieved 2008-01-13.
[5] Peck, Morgen, Equality of Mathematicians (http://scienceline.org/2007/01/31/math_controversy_peck/), , "Alexandre Grothendieck is

arguably the most important mathematician of the 20th century..."
[6] Leith, Sam (20 March 2004), "The Einstein of maths" (http://www.lewrockwell.com/spectator/spec262.html), The Spectator, , "[A]

mathematician of staggering accomplishment ... a legendary figure in the mathematical world."
[7] at p. 2. (http://www.ams.org/notices/200309/rev-raynaud.pdf,)

[8] Society for Industrial and Applied Mathematics (http://www.siam. org/news/news.php?id=1405)

[9] Allyn Jackson, The Life of Alexander Grothendieck, p. 1040 (http://www.ams.org/notices/200409/fea-grothendieck-partl.pdf)
[10] See Jackson (2004: 1).

[II] Crafoord Prize letter (http://web.archive.Org/web/20060106062005/http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf)
[12] Matthews, Robert (20 August 2006). "Mathematics, where nothing is ever as simple as it seems" (http://www.telegraph.co.uk/news/

1526781/Mathematics-where-nothing-is-ever-as-simple-as-it-seems.html). Daily Telegraph. . Retrieved 5 July 2009.
[13] The editors were Pierre Cartier, Luc Illusie, Nick Katz, Gerard Laumon, Yuri Manin, and Ken Ribet. A second edition has been printed

(2007) by Birkhauser.
[14] The IHES at Forty (http://www.ams.org/notices/199903/ihes-changes.pdf) Allyn Jackson, March 1999, Noticed of the AMS pp.

329-337
[15] Google book link (http://books. google. co.uk/books?id=Hlflj2XmXkcC&dq=Geometric+Galois+Actions&printsec=frontcover&

source=bl&ots=q9iN4QMkYj&sig=fAsvVlPOhN9K9LfjGFJC5zm_IA4&hl=en&ei=KW-3ScSkHYHIMuDukOEK&sa=X&

oi=book_result&resnum=l&ct=result)
[16] http://www.math.jussieu.fr/~leila/grothendieckcircle/index.php

[17] ESQUISSE D'UN PROGRAMME par Alexandre Grothendieck (http://matematicas.unex.es/~navarro/res/esquissefr.pdf)
[18] http://www.bangor.ac.Uk/r.brown

[19] In Russian (http://www.mccme.ru/free-books/grothendieck/RS.html)
[20] COSECHAS Y SIEMBRAS: Reflexiones y testimonios sobre un pasado de matem'atico (http://matematicas.unex.es/~navarro/res/

preludio.pdf) Preludio
[21] COSECHAS Y SIEMBRAS: Reflexiones y testimonios sobre un pasado de matem'atico (http://matematicas.unex.es/~navarro/res/carta.

pdf) Carta
[22] La Clef des Songes (http://matematicas.unex.es/~navarro/res/clefl-6.pdf)
[23] http://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter

References

• Cartier, Pierre (1998), "La folle journee, de Grothendieck a Connes et Kontsevich — Evolution des notions
d'espace et de symetrie", Les relations entre les mathematiques et la physique theorique — Festschrift for the
40th anniversary of the IHES, Institut des Hautes Etudes Scientifiques, pp. 11—19

• Cartier, Pierre (2001), "A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of
concepts of space and symmetry" (http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/
S0273-0979-01-00913-2.pdf) (PDF), Bull. Amer. Math. Soc. 38 (4): 389-408,

doi: 10. 1090/S0273-0979-0 1-009 13-2. An English translation of Cartier (1998)

• Deligne, Pierre (1998), "Quelques idees mattresses de l'oeuvre de A. Grothendieck" (http://smf.emath.fr/
Publications/SeminairesCongres/1998/3/pdf/smf_sem-cong_3_l l-19.pdf), Materiaux pour I'histoire des
mathematiques au XXe siecle —Actes du colloque a la memoire de Jean Dieudonne (Nice 1996), Societe
Mathematique de France, pp. 1 1—19

• Dieudonne, Jean Alexandre (1990), "De L'analyse fonctionelle aux fondements de la geometrie algebrique", in
Cartier, Pierre et al., The Grothendieck Festschrift, Volume 1, Birkhauser, pp. 1-14, ISBN 978-0-8176-4566-3

• Jackson, Allyn (2004), "Comme Appele du Neant — As If Summoned from the Void: The Life of Alexandre
Grothendieck I" (http://www.ams.org/notices/200409/fea-grothendieck-partl.pdf) (PDF), Notices of the
American Mathematical Society 51 (4): 1038—1056

Alexander Grothendieck 452

• Jackson, Allyn (2004), "Comme Appele du Neant — As If Summoned from the Void: The Life of Alexandre
Grothendieck II" (http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf) (PDF), Notices of the
American Mathematical Society 51 (10): 1196—1212

• Rehmeyer, Julie (9 May 2008), "Sensitivity to the Harmony of Things" (http://www.sciencenews.org/view/
generic/id/3 1 898/title/Sensitivity_to_the_harmony_of_things), Science News

• Scharlau, Winfred, Wer ist Alexander Grothendieck? : Anarchie,Mathematik, Spiritualitat (http://www.
scharlau-online.de/ag_l.html) Three-volume biography.

• Scharlau, Winifred (September 2008), written at Oberwolfach, Germany, "Who is Alexander Grothendieck"
(http://www.ams.org/notices/200808/tx080800930p.pdf) (PDF), Notices of the American Mathematical
Society (Providence, RI: American Mathematical Society) 55 (8): 930-941, ISSN 1088-9477, OCLC 34550461,
retrieved 2008-09-02

This article incorporates material from Alexander Grothendieck (http:/ / planetmath. org/ encyclopedia/
AlexanderGrothendieck.html), which is licensed under the Creative Commons Attribution/Share-Alike License.

External links

• O'Connor, John J.; Robertson, Edmund F., "Alexander Grothendieck" (http://www-history.mcs.st-andrews.ac.
uk/Biographies/Grothendieck.html), MacTutor History of Mathematics archive, University of St Andrews.

• Alexander Grothendieck (http://www. genealogy. ams.org/id.php?id=31245) at the Mathematics Genealogy
Project

• Grothendieck Circle (http://www.grothendieckcircle.org/), collection of mathematical and biographical
information, photos, links to his writings

• Institut des Hautes Etudes Scientifiques (http://www.ihes.fr)

• The origins of "Pursuing Stacks' (http://www.bangor.ac.Uk/r.brown/pstacks.htm) This is an account of how
"Pursuing Stacks' was written in response to a correspondence in English with Ronnie Brown and Tim Porter
(http://www.bangor.ac.uk/~mas013/) at Bangor, which continued until 1991.

• Recoltes et Semailles (http://acm.math.spbu.ru/RS/) in French.

• Spanish translation (http://matematicas.unex.es/~navarro/res/) of "Recoltes et Semailles" et "Le Clef des
Songes" and other Grothendieck's texts

• short bio (http://www.ams.org/notices/200808/tx080800930p.pdf) from Notices of the American
Mathematical Society

Charles Ehresmann 453

Charles Ehresmann

Charles Ehresmann (right) at the topology conference 1949 in Oberwolfach, together with Paul Vincensini (middle) and Georges Reeb (left)

Born 19 April 1905StraGburg, Alsace-Lorraine, German Empire (today Strasbourg,

Alsace, France)

Died 22 September 1979Amiens, Picardy, France

Fields Mathematics

Alma mater Ecole Normale Superieure

Doctoral advisor Elie Cartan

Doctoral students Georges Reeb

Wu Wen-Tsiin
Andre Haefliger
Valentin Poenaru
Daniel Tanre

Known for Ehresmann's theorem

Ehresmann connection

Charles Ehresmann (1905-1979) was a French mathematician who worked on differential topology and category
theory. He is known for work on the topology of Lie groups, the jet concept (see jet bundle), and his seminar on
category theory.

He attended the Ecole Normale Superieure in Paris before performing one year of military service. He finished his
PhD thesis Sur la topologie de certains espaces homogenes (French: On the topology of certain homogeneous
spaces) in 1934 under the supervision of Elie Cartan.

In 1957 he founded the mathematical journal Cahiers de Topologie et Geometrie Differentielle Categoriques.

Jean Dieudonne describes Ehresmann's personality as "... distinguished by forthrightness, simplicity, and total
absence of conceit or careerism. As a teacher he was outstanding, not so much for the brilliance of his lectures as
for the inspiration and tireless guidance he generously gave to his research students ... "

He had 76 Ph.D. students, including Georges Reeb, Wu Wen-Tsiin, Andre Haefliger, Valentin Poenaru, Daniel
Tame.

Charles Ehresmann 454

References

• International Conference "Charles Ehresmann : 100 ans", Universite de Picardie Jules Verne a Amiens, 7-8-9
October 2005, http://pagesperso-orange.fr/vbm-ehr/ChEh/indexAng.htm

• "The mathematical legacy of Charles Ehresmann', Proceedings of the 7th Conference on the Geometry and
Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland)
8.05.2005-15.05.2005, Edited by J. Krysinski, J. Pradines, T. Rybicki, R. Wolak, Banach Centre Publications 76,
Institute of Mathematics Polish Academy of Sciences, Warsaw, (2007).

External links

• O'Connor, John J.; Robertson, Edmund F., "Charles Ehresmann" , MacTutor History of Mathematics archive,
University of St Andrews.

• Charles Ehresmann at the Mathematics Genealogy Project

References

[1] http://www-history.mcs.st-andrews.ac.uk/Biographies/Ehresmann.html
[2] http://www. genealogy. ams.org/id.php?id=96080

Samuel Eilenberg

455

Samuel Eilenberg

Samuel Eilenberg (1970)

Born

September 30, 1913Warsaw, Russian

Empire

Died

January 30, 1998 (aged 84)New York

City, New York

Nationality

Polish American

Fields

Mathematics

Institutions

Columbia University

Alma mater

University of Warsaw

Doctoral advisor

Kazimierz Kuratowski
Karol Borsuk

Doctoral students

David Buchsbaum
Alex Heller
Daniel Kan
William Lawvere
Ramaiyengar Sridharan

Known for

Eilenberg— Steenrod axioms
Eilenberg swindle

Samuel Eilenberg (September 30, 1913 — January 30, 1998) was a Polish and American mathematician of Jewish
descent. He was born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had
spent much of his career as a professor at Columbia University.

He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor was Karol Borsuk. His main interest was
algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names
the Eilenberg— Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process,
Eilenberg and Mac Lane created category theory.

Eilenberg was a member of the Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which
became a classic.

Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle
(or telescope) is a construction applying the telescoping cancellation idea to projective modules.

Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced
by Eilenberg in 1974.

Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of small sculptures and other
artifacts from India, Indonesia, Pakistan, Nepal, Thailand, Cambodia, Sri Lanka and Central Asia. In 1991-1992, the
Metropolitan Museum of Art in New York staged an exhibition from more than 400 items that Eilenberg had

Samuel Eilenberg 456

donated to the museum, entitled The Lotus Transcendent: Indian and Southeast Asian Art From the Samuel
Eilenberg Collection".

Selected publications

• Samuel Eilenberg, Automata, Languages and Machines. ISBN 0-12-234001-9.

• Samuel Eilenberg & Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups , Annals of
Mathematics, 2nd Ser., 65 (1957), no. 3, 517 - 518. MR0085510

• Samuel Eilenberg & Saunders Mac Lane, "Relations between homology and homotopy groups of spaces", Annals
of Mathematics 46 (1945), 480-509.

• Samuel Eilenberg & Saunders Mac Lane, "Relations between homology and homotopy groups of spaces. II",
Annals of Mathematics 51 (1950), 514-533.

• Eilenberg, Samuel; Moore, John C. (1962), "Limits and spectral sequences", Topology 1 (1): 1—23,
doi: 10. 1016/0040-9383(62)90093-9, ISSN 0040-9383

• Samuel Eilenberg & Norman E. Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. U. S. A.
31,(1945). 117—120.

• Samuel Eilenberg & Norman E. Steenrod, Foundations of algebraic topology, Princeton University Press,
Princeton, New Jersey, 1952. xv+328 pp.

Footnotes

[1] New York Times obituary, February 3, 1998.

[2] http://links.jstor.org/sici?sici=0003-486X%28195705%292%3A65%3A3%3C517%3AOTLCOA%3E2.0.CO%3B2-J

External links

• Samuel Eilenberg (http://www. genealogy. ams.org/id.php?id=7643) at the Mathematics Genealogy Project

• O'Connor, John J.; Robertson, Edmund F., "Samuel Eilenberg" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Eilenberg.html), MacTutor History of Mathematics archive, University of St Andrews.

• Eilenberg's biography (http://newton.nap.edu/html/biomems/seilenberg.html) - from the National
Academies Press, by Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller and Saunders Mac Lane.

Emil Artin

457

Emil Artin

Born
Died

Fields

Institutions

Alma mater

Emil Artin

March 3, 1898 Vienna, Austria

December 20, 1962 (aged 64)Hamburg, Germany

Mathematics

University of Hamburg
University of Notre Dame
Indiana University
Princeton University

University of Vienna
University of Leipzig

Doctoral advisor Gustav Herglotz

Otto Ludwig Holder

Doctoral students Bernard Dwork
Serge Lang

Kollagunta Ramanathan
John Tate
Hans Zassenhaus
Max Zorn

Emil Artin (March 3, 1898, in Vienna
mathematician.

December 20, 1962, in Hamburg) was an Austrian- Armenian

Biography
Parents

The mathematician Emil Artin was born on March 3, 1898 in Vienna to parents Emma Maria, nee Laura (stage name
Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin,
Austrian-born of Armenian descent. Several documents, including Emil's birth certificate, list the father's occupation
as "opera singer" though others list it as "art dealer." It seems at least plausible that he and Emma had met as
colleagues in the theater. They had been married in St. Stephen's Parish on July 24, 1895.

Emil Artin 458

Early education

Emil entered school in September 1904, presumably in Vienna. By then, his father was already suffering symptoms
of advanced syphilis, among them increasing mental instability, and was eventually institutionalized at the recently
established (and imperially sponsored) insane asylum at Mauer Ohling, 125 kilometers west of Vienna. It is notable
that neither wife nor child contracted this highly infectious disease. The senior Emil Artin died there July 20, 1906.
Young Emil was eight.

On July 15, 1907, Emil's mother remarried — her second husband, Rudolf Hubner a prosperous manufacturer in the
German-speaking city of Reichenberg, Czechoslovakia (now Liberec, in the Czech Republic). Documentary
evidence suggests that Emma had already been resident in Reichenberg the previous year, and in deference to her
new husband, she had abandoned her vocal career. Hubner deemed a life in the theater unseemly in the wife of a man
of his position.

In September, 1907, Emil entered the Volksschule in Strobnitz, a small town in southern Czechoslovakia near the
Austrian border. For that year, he lived away from home, boarding on a local farm. The following year, he returned
to the home of his mother and stepfather, and entered the Realschule in Reichenberg, where he pursued his
secondary education to June, 1916.

In Reichenberg, Emil formed a life-long friendship with a young neighbor, Arthur Baer, who became an astronomer,
teaching for many years at Cambridge University. Astronomy was an interest the two boys shared already at this
time. They each had telescopes. They also rigged a telegraph between their houses, over which once Baer excitedly
reported to his friend an astronomical discovery he thought he had made — perhaps a supernova, he thought — and
told Emil where in the sky to look. Emil tapped back the terse reply "A-N-D-R-O-M-E-D-A N-E-B-E-L."
(Andromeda nebula)

Emil's academic performance in the first years at the Realschule was spotty. Up to the end of the 1911—1912 school
year, for instance, his grade in mathematics was merely "geniigend," (satisfactory). Of his mathematical inclinations
at this early period he later wrote, "Meine eigene Vorliebe zur Mathematik zeigte sich erst im sechzehnten
Lebensjahr, wahrend vorher von irgendeiner Anlage dazu uberhaupt nicht die Rede sein konnte." ("My own
predilection for mathematics manifested itself only in my sixteenth year, whereas earlier there was absolutely no
question of any particular aptitude for it.") His grade in French for 1912 was actually "nicht geniigend"
(unsatisfactory). He did rather better work in physics and chemistry. But from 1910 to 1912, his grade for
"Comportment" was "nicht geniigend."

Emil spent the school year 1912—1913 away from home, in France, a period he spoke of later as one of the happiest
of his life. He lived that year with the family of Edmond Fritz, in the vicinity of Paris, and attended a school there.
When he returned from France to Reichenberg, his academic work markedly improved, and he began consistently
receiving grades of "gut" or "sehr gut" (good or very good) in virtually all subjects — including French and
"Comportment." By the time he completed studies at the Realschule in June, 1916, he was awarded the Reifezeugnis
(diploma — not to be confused with the Abitur) that affirmed him "reif mit Auszeichnung" (qualified with distinction)
for graduation to a technical university.

Emil Artin 459

University education

Now that it was time to move on to university studies, Emil was no doubt content to leave Reichenberg, for relations
with his stepfather were clouded. According to him, Hilbner reproached him "day and night" as a financial burden,
and even when Emil became a university lecturer and then a professor, Hilbner deprecated his academic career as
self-indulgent and belittled its paltry remuneration.

In October, 1916, Emil matriculated at the University of Vienna, having focused by now on mathematics. He studied
there with Phillip Furtwangler, and also took courses in astrophysics and Latin.

Studies at Vienna were interrupted when Emil was drafted in June, 1918 into the Austrian army (his Army photo ID
is dated July 1, 1918). Assigned to the K.u. K. 44th Infantry Regiment, he was stationed northwest of Venice at
Primolano, on the Italian front in the foothills of the Dolomites. To his great relief, Emil managed to avoid combat
by volunteering for service as a translator — his ignorance of Italian notwithstanding. He did know French, of course,
and some Latin, was generally a quick study, and was motivated by a highly rational fear in a theater of that war that
had all too often proven a meat-grinder. In his scramble to learn at least some Italian, Emil had recourse to an
encyclopedia, which he once consulted for help in controlling the cockroaches infesting the Austrian barracks. At
some length, the article described a variety of elaborate methods, concluding finally with — Emil laughingly recalled
in later years — "la caccia diretta" ("the direct hunt"). Indeed, "la caccia diretta" was the straight-forward method he
and his fellow infantrymen adopted.

Emil survived both war and vermin on the Italian front, and returned late in 1918 to the University of Vienna, where
he remained through Easter of the following year.

By June 1919, he had moved to Leipzig and matriculated at the University there as a "Class 2 Auditor" ("Horer
zweiter Ordnung"). Late the same year, Emil undertook the formality of standing for a qualifying examination by an
academic board of the Oberrealschule in Leipzig, which he passed with the grade of "gut" (good), receiving for the
second time the Reifezeugnis (diploma attesting the equivalence of satisfactory completion of 6 years at a
Realschule.) How this Leipzig Reifezeugnis differed technically from the one he had been granted at Reichenberg is
unclear from the document, but it apparently qualified him to matriculate as a regular student at the University,
which normally required the Abitur.

From 1919 to June 1921, Emil pursued mostly mathematical studies at Leipzig. His principal teacher and dissertation
advisor was Gustav Herglotz. Additionally, Emil took courses in chemistry and various fields of physics, including
mechanics, atomic theory, quantum theory, Maxwellian theory, radioactivity, and astrophysics. In June, 1921 he was
awarded the Doctor of Philosophy degree, based on his "excellent" dissertation, "Quadratische Korper im Gebiete der
hoheren Kongruenzen" ("On the Arithmetic of Quadratic Function Fields over Finite Fields"), and the oral
examination which — his diploma affirms — he had passed three days earlier "with extraordinary success."

In the fall of 1921, Emil moved to Gottingen, considered the Mecca of mathematics at the time, where he pursued
one year of post-doctoral studies in mathematics and mathematical physics with Richard Courant and David Hilbert.
While at Gottingen, he worked closely with Emmy Noether and Helmut Hasse.

Aside from consistently good school grades in singing, the first documentary evidence of Emil's deep and life-long
engagement with music comes from the year in Gottingen, where he was regularly invited to join in the chamber
music sessions hosted by Richard Courant. He played all the keyboard instruments, and was an especially
accomplished flautist, although it is not known exactly by what instruction he had achieved proficiency on these
instruments. He became especially devoted to the music of J. S. Bach.

Emil Artin 460

Professorship at Hamburg

Courant arranged for Emil to receive a stipend for the summer of 1922 in Gottingen, which occasioned his declining
a position offered him at the University of Kiel. The following October, however, he accepted an equivalent position
at Hamburg, where in 1923, he completed the Habilitation thesis (required of aspirants to a professorship in
Germany), and on July 24 advanced to the rank of Privatdozent.

On April 1, 1925, Emil was promoted to Associate Professor (auBerordentlicher Professor). In this year also, Emil
applied for and was granted German citizenship. And on October 15, 1926, he was promoted to full Professor
(ordentlicher Professor).

Early in the summer of 1925, Emil attended the Congress of the Wandervogel youth movement at Wilhelmshausen
near Kassel with the intention of gathering a congenial group to undertake a trek through Iceland later that summer.
Iceland (before the transforming presence of American and British forces stationed there during WWII) was still a
primitive country in 1925, with a thinly scattered population and little transportation infrastructure. Emil succeeded
in finding six young men to join him in this adventure. In the second half of August, 1925, the group set out by
steamer from Hamburg, first to Norway, where they boarded a second steamer that took them to Iceland, stopping in
several of the east fjords before arriving at their destination, Husavik in the north of the island. Here the
Wandervogel group disembarked, their initial goal trekking down the Laxa River to Lake Myvatn. They made a
circuit of the large, irregular lake, staying in farm houses, barns, and occasionally a tent as they went. When they
slept in barns, it was often on piles of wet straw or hay. On those lucky occasions when they slept in beds, it could be
nearly as damp on account of the rain trickling through the sod roofs. The tent leaked as well.

Emil kept a meticulous journal of this trip, making daily entries in a neat, minuscule hand. He and several of the
young men had brought cameras, so that the trek is documented also by nearly 200 small photographs. Emil's journal
attests to his over-arching interest in the geology of this mid-Atlantic island, situated over the boundary of two
tectonic plates whose shifting relation makes it geologically hyperactive.

In keeping with the Wandervogel ethos, Emil and his companions carried music with them wherever they visited.
The young men had packed guitars and violins, and Emil played the harmoniums common in the isolated farmsteads
where they found lodging. The group regularly entertained their Icelandic hosts, not in full exchange for board and
lodging, to be sure, but for goodwill certainly, and occasioning sometimes even a little extra on their plates and a
modestly discounted tariff.

From Lake Myvatn, Emil and his companions headed west towards Akureyri, passing the large waterfall GoSafoss
on the way. From Akureyri, they trekked west down the Oxnadalur (Ox Valley) intending to rent pack horses and
cross the high and barren interior by foot to Reykjavik. By the time they reached the lower end of Skagafjorflur,
however, they were persuaded by a local farmer from whom they had hoped to rent the horses that a cross-country
trek was by then impracticable since, with the approach of winter, highland routes were already snow-bound and
impassable. Instead of turning south, then, they turned north to SiglufjorSur, where they boarded another steamer
that took them around the western peninsula and down the coast to Reykjavik. From Reykjavik, they returned via
Norway to Hamburg. By Emil's calculation the distance they had covered on foot through Iceland totaled 450
kilometers.

Early in 1926, the University of Miinster offered Emil a professorial position; however, Hamburg matched the offer
financially, and (as noted above) promoted him to full professor, making him (along with his young colleague
Helmut Hasse) one of the two youngest professors of mathematics in Germany.

It was in this period that he acquired his lifelong nickname, "Ma," short for mathematics, which he came to prefer to
his given name, and by which virtually everyone who knew him well called him. Although the nickname might seem
to imply a narrow intellectual focus, quite the reverse was true of Emil. Even his teaching at the University of
Hamburg went beyond the strict boundaries of mathematics to include mechanics and relativity theory. He kept up
on a serious level with advances in astronomy, chemistry and biology (he owned and used a fine microscope), and
the circle of his friends in Hamburg attests to the catholicity of his interests. It included the painter Heinrich

EmilArtin 461

Stegemann, and the author and organ-builder Hans Henny Jahn. Stegemann was a particularly close friend, and made
portraits of Emil, Natascha and the two children born in Hamburg. Music continued to play a central role in his life;
he acquired a Neupert double manual harpsichord, and a clavichord made by the Hamburg builder Walter Ebeloe, as
well as a silver flute made in Hamburg by G. Urban. Chamber music gatherings became a regular event at the Artin
apartment as they had been at the Courants in Gottingen.

On August 15, 1929, Emil married Natalia Naumovna Jasny (Natascha), a young Russian emigre who had been a
student in several of his classes. One of their shared interests was photography, and when Emil bought a Leica for
their joint use (a Leica A, the first commercial model of this legendary camera), Natascha began chronicling the life
of the family, as well as the city of Hamburg. For the next decade, she made a series of artful and expressive portraits
of Emil that remain by far the best images of him taken at any age. Emil, in turn, took many fine and evocative
portraits of Natascha. Lacking access to a professional darkroom, their films and prints had to be developed in a
makeshift darkroom set up each time (and then dismantled again) in the small bathroom of whatever apartment they
were occupying. The makeshift darkroom notwithstanding, the high artistic level of the resulting photographic prints
is attested to by the exhibit of Natascha's photographs mounted in 2001 by the Museum fur Kunst und Gewerbe
Hamburg, and its accompanying catalogue, "Hamburg — Wie Ich Es Sah."

In 1930, Emil was offered a professorship at ETH (Eidgenossische Technische Hochschule) in Zurich, to replace
Hermann Weyl, who had moved to Gottingen. He chose to remain at Hamburg, however. Two years later, in 1932,
for contributions leading to the advancement of mathematics, Emil was honored — jointly with Emmy Noether — with
the award of the Alfred Ackermann-Teubner Memorial Prize, which carried a grant of 500 marks.

Nazi period

In January 1933 — a tragically fateful month in German history — Natascha gave birth to their first child, Karin. A
year and a half later, in the summer of 1934, son Michael was born. The political climate at Hamburg was not so
poisonous as that at Gottingen, where by 1935 the mathematics department had been purged of Jewish and dissident
professors. Still, Emil's situation became increasingly precarious, not only because Natascha was half Jewish, but
also because Emil made no secret of his distaste for the Hitler regime. At one point Blaschke, by then a Nazi Party
member, but nonetheless solicitous of the Artins' well-being, warned Emil discreetly to close his classroom door so
his frankly anti-Nazi comments couldn't be heard by passersby in the hallway.

Natascha recalled going down to the newsstand on the corner one day and being warned in hushed tones by the man
from whom she and Emil bought their paper that a man had daily been watching their apartment from across the
street. Once tipped off, she and Emil became very aware of the watcher (Natascha liked to refer to him as their
"spy"), and even rather enjoyed the idea of his being forced to follow them on the long walks they loved taking in the
afternoons to a cafe far out in the countryside.

Toying with their watcher on a fine Autumn afternoon was one thing, but the atmosphere was in fact growing
inexorably serious. Natascha's Jewish father and her sister, seeing the handwriting on the wall, had already left for
the U.S. in the summer of 1933. As half- Jewish, Natascha's status was, if not ultimately quite hopeless, certainly not
good. Hasse, like Blaschke a nationalistic supporter of the regime, had applied for Party membership, but was
nonetheless no anti-Semite. Besides he was a long-time friend and colleague of Emil's. He suggested that the two
Artin children — only one quarter Jewish, or in Nazi terminology, "Mischlinge zweiten Grades" — might, if a few
strategic strings could be pulled, be officially "aryanized." Hasse offered to exert his influence with the Ministry of
Education (Kultur- und Schulbehorde, Hochschulwesen), and Emil — not daring to leave any stone unturned,
especially with respect to the safety of his children — went along with this effort. He asked his father-in-law, by then
resident in Washington D.C., to draft and have notarized an affidavit attesting to the Christian lineage of his late
wife, Natascha's mother. Emil submitted this affidavit to the Ministry of Education, but to no avail.

By this time, to be precise, on July 15, 1937, because of Natascha's status as "Mischling ersten Grades," Emil had
lost his post at the University — technically, compelled into early retirement — on the grounds of paragraph 6 of the

Emil Artin 462

Act to Restore the Professional Civil Service (Gesetz zur Wiederherstellung des Berufsbeamtentums) of April 7,
1933. Ironically, he had applied only some months earlier, on February 8, 1937, for a leave of absence from the
University in order to accept a position offered him at Stanford. On March 15, 1937, the response had come back
denying his application for leave on the grounds that his services to the University were indispensable ("Da die
Tatigkeit des Professors Dr. Artin an der Universitat Hamburg nicht entbehrt werden kann. . .")

By July, when he was summarily "retired," ("in Ruhestand versetzt") the position at Stanford had been filled.
However, through the efforts of Richard Courant (by then in New York), and Solomon Lefschetz at Princeton, a
position was found for him at Notre Dame University in South Bend, Indiana.

Emigration to the U.S.

The family must have worked feverishly to prepare for emigration to the United States, for this entailed among other
things packing their entire household for shipment. Since German law forbade emigrants taking more than a token
sum of money out of the country, the Artins sank all the funds at their disposal into shipping their entire household,
from beds, tables, chairs and double-manual harpsichord down to the last kitchen knife, cucumber sheer, and potato
masher to their new home. This is why each of their residences in the United States bore such a striking resemblance
to the rooms photographed so beautifully by Natascha in their Hamburg apartment (see Natascha A. Brunswick,
"Hamburg: Wie Ich Es Sah," Dokumente der Photographie 6, Museum fur Kunst und Gewerbe Hamburg, 2001,
pp. 48-53) .

On the morning they were to board the Hamburg-Amerika line ship in Bremerhaven, October 21, 1937, daughter
Karin woke with a high temperature. Terrified that should this opportunity be missed, the window of escape from
Nazi Germany might close forever, Emil and Natascha chose to risk somehow getting Karin past emigration and
customs officials without their noticing her condition. Anxiously, they managed to conceal Karin's feverish state,
and without incident boarded the ship, as many left behind were tragically never able to do. When they landed a
week later at Hoboken, Richard Courant and Natascha's father, the Russian agronomist Naum Jasny (then working
for the U.S. Department of Agriculture) were on the dock to welcome the family to the United States.

Bloomington years

It was early November, 1937 by the time they arrived in South Bend, where Emil joined the faculty at Notre Dame,
and taught for the rest of that academic year. He was offered a permanent position the following year 170 miles to
the south at Indiana University, in Bloomington. Shortly after the family resettled there, a second son, Thomas, was
born on November 12, 1938.

After moving to Bloomington, Emil quickly acquired a piano, and soon after that a Hammond Organ, a recently
invented electronic instrument that simulated the sound of a pipe organ. He wanted this instrument in order primarily
to play the works of J. S. Bach, and because the pedal set that came with the production model had a range of only
two octaves (not quite wide enough for all the Bach pieces), he set about extending its range. Music was a constant
presence in the Artin household. Karin played the cello, and then the piano as well, and Michael played the violin.
As in Hamburg, the Artin living room was regularly the venue for amateur chamber music performances.

The circle of the Artins' University friends reflected Emil's wide cultural and intellectual interests. Notable among
them were Alfred Kinsey and his wife of the Psychology Department, as well as prominent members of the Fine
Arts, Art History, Anthropology, German Literature, and Music Departments. For several summer semesters, Emil
accepted teaching positions at other universities, viz., Stanford in 1939 and 1940, The University of Michigan at Ann
Arbor in 1941 and 1951, and The University of Colorado, in Boulder, in 1953. On each of these occasions, the
family accompanied him.

Emil insisted that only German be spoken in the house. Even Tom, born in the U.S., spoke German as his first
language, acquiring English only from his siblings and his playmates in the neighborhood; for the first four or five
years of his life, he spoke English with a pronounced German accent. Consistent with his program of maintaining the

Emil Artin 463

family's German cultural heritage, Emil gave high priority to regularly reading German literature aloud to the
children. The text was frequently from Goethe's autobiographical "Dichtung und Wahrheit," or his poems,
"Erlkonig," for instance. Occasionally, he would read from an English text. Favorites were Mark Twain's "Tom
Sawyer," Charles Dickens's "A Christmas Carol," and Oscar Wilde's "The Canterville Ghost." For the Artin children,
these readings replaced radio entertainment, which was strictly banned from the house. There was a radio, but (with
the notable exception of Sunday morning broadcasts by E. Power Biggs from the organ at the Busch-Reisinger
Museum in Cambridge, to which Emil and Natascha listened still lounging in bed) it was switched on only to hear
news of the war. Similarly, the Artin household would never in years to come harbor a television set. Once the war
had ended, the radio was retired to the rear of a dark closet.

As German citizens, Emil and Natascha were technically classified as enemy aliens for the duration of the war. On
April 12, 1945, with the end of the war in Europe only weeks away, they applied for naturalization as American
citizens. American citizenship was granted them on February 7, 1946.

On the orders of a Hamburg doctor whom he had consulted about a chronic cough, Emil had given up smoking years
before. He had vowed not to smoke so long as Hitler remained in power. On May 8, 1945, at the news of Germany's
surrender and the fall of the Third Reich, Natascha made the mistake of reminding him of this vow, and in lieu of a
champagne toast, he indulged in what was to be the smoking of a single cigarette. Unfortunately, the single cigarette
led to a second, and another after that. Emil returned to heavy smoking for the rest of his life, a habit that
unquestionably contributed to his premature death.

Princeton years

If Gottingen had been the "Mecca" of mathematics in the 1920s and early '30's, Princeton, following the decimation
of German mathematics under the Nazis, had become the center of the mathematical world in the 1940's. In April,
1946, Emil was appointed Professor at Princeton, at a yearly salary of $8,000. The family moved there in the fall of
1946.

Notable among his graduate students at Princeton are Serge Lang, John Tate, and Timothy O'Meara. But Emil chose
also to teach the honors section of Freshman calculus each year. He was renowned for the elegance of his teaching.
Frei and Roquette write that Artin's "main medium of communication was teaching and conversation: in groups,
seminars and in smaller circles. We have many statements of people near to him describing his unpretentious way of
communicating with everybody, demanding quick grasp of the essentials but never tired of explaining the necessary.
He was open to all kinds of suggestions, and distributed joyfully what he knew. He liked to teach, also to young
students, and his excellent lectures, always well prepared but without written notes, were hailed for their clarity and
beauty." (Emil Artin and Helmut Hasse: Their Correspondence 1923—1934, Introduction.)

Whenever he was asked whether mathematics was a science, Emil would reply unhesitatingly, "No. An art." His
elegant elaboration of this idea is often cited, and worth repeating here: "We all believe that mathematics is an art.
The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers,
to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, each conclusion is drawn from
previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception
should be instantaneous. We have all experienced on some rare occasion the feeling of elation in realizing that we
have enabled our listeners to see at a glance the whole architecture and all its ramifications."

It has even been said — only half in jest — that his lectures could be too perfect, lulling a hearer into believing he had
understood and assimilated an idea or a proof which, on waking the following day might seem as remote and
chimerical as ever.

During the Princeton years, Emil built a 6 inch reflecting telescope to plans he found in "Sky and Telescope"
magazine, to which he subscribed. He spent weeks in the basement attempting to grind the mirror to specifications,
without success, and his continued failure to get it right led to increasing frustration. Then, in California to give a
talk, he made a side trip to the Mt. Wilson Observatory, where he discussed his project with the astronomers.

Emil Artin 464

Whether it was their technical advice, or Natascha's intuitive suggestion that it might be too cold in the basement,
and that he should try the procedure upstairs in the warmth of his study (which he did), he completed the grinding of
the mirror in a matter of days. With this telescope, he surveyed the night skies over Princeton.

In September 1955, Emil accepted an invitation to visit Japan. From his letters, it is clear he was treated like royalty
by the Japanese mathematical community, and was charmed by the country. A confirmed atheist most of his life, he
was nonetheless interested in learning about the diverse threads of Buddhism, and visiting its holy sites. In a letter
home he describes his visit to the temples at Nara. "Then we were driven to a place nearby, Horiuji [Horyu-ji] where
a very beautiful Buddhist temple is. We were received by the abbot, and a priest translated into English. We obtained
the first sensible explanation about modern Buddhism. The difficulty of obtaining such an explanation is enormous.
To begin with most Japanese do not know and do not understand our questions. All this is made more complicated
by the fact that there are numerous sects and each one has another theory. Since you get your information only piece
wise, you cannot put it together. This results in an absurd picture. I am talking of the present day, not of its original
form."

His letter goes on to outline at length the general eschatological framework of Buddhist belief. Then he adds, "By the
way, a problem given by the Zens for meditation is the following: If you clap your hands, does the sound come from
the left hand or from the right?"

Return to Hamburg

The following year, Emil took a leave of absence to return to Germany for the first time since emigration, nearly
twenty years earlier. He spent the fall semester at Gottingen, and the next at Hamburg. For the Christmas holidays,
he travelled to his birthplace, Vienna, to visit his mother, a city he had not seen in decades. In a letter home he
described the experience of his return in a single, oddly laconic sentence: "It is kind of amusing to walk through
Vienna again." In 1957, an honorary doctorate was conferred on Emil by the University of Freiburg. That fall, he
returned to Princeton for what would be his final academic year at that institution.

Emil's marriage to Natascha had by this time seriously frayed. Though nominally still husband and wife, resident in
the same house, they were for all intents and purposes living separate lives. Emil was offered a professorship at
Hamburg, and at the conclusion of Princeton's spring semester, 1958, he moved permanently to Germany. His
decision to leave Princeton University and the United States was complicated, based on multiple factors, prominent
among them Princeton's (then operative) mandatory retirement age of 65. Emil had no wish to retire from teaching
and direct involvement with students. Hamburg's offer was open-ended.

Emil and Natascha were divorced in 1959. In Hamburg, Emil had taken an apartment, but soon gave it over to his
mother whom he had brought from Vienna to live near him in Hamburg. He in turn moved in with Hel Braun into an
apartment in the same neighborhood. On January 4, 1961, he was granted German citizenship. In June, 1962, on the
occasion of the 300th anniversary of the death Blaise Pascal, the University of Clermont-Ferrand conferred an
honorary doctorate on him. On December 20 of the same year, Emil Artin died at home in Hamburg, aged 64, of a
heart attack.

The University of Hamburg honored his memory on April 26, 2005 by naming one of its newly renovated lecture
halls The Emil Artin Lecture Hall.

Emil Artin 465

Influence and work

He was one of the leading algebraists of the century, with an influence larger than might be guessed from the one
volume of his Collected Papers edited by Serge Lang and John Tate. He worked in algebraic number theory,
contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure
theories of rings, groups and fields. He developed the theory of braids as a branch of algebraic topology.

He was also an important expositor of Galois theory, and of the group cohomology approach to class ring theory
(with John Tate), to mention two theories where his formulations became standard. The influential treatment of
abstract algebra by van der Waerden is said to derive in part from Artin's ideas, as well as those of Emmy Noether.
He wrote a book on geometric algebra that gave rise to the contemporary use of the term, reviving it from the work
ofW. K.Clifford.

Conjectures

He left two conjectures, both known as Artin's conjecture. The first concerns Artin L-functions for a linear
representation of a Galois group; and the second the frequency with which a given integer a is a primitive root
modulo primes p, when a is fixed and p varies. These are unproven; Hooley proved a result for the second
conditional on the first.

Supervision of research

Artin advised over thirty doctoral students, including Bernard Dwork, Serge Lang, K. G Ramanathan, John Tate,
Hans Zassenhaus and Max Zorn. A more complete list of his students can be found at the Mathematics Genealogy
Project website (see "External Links," below).

Family

In 1932 he married Natascha Jasny, born in Russia to mixed parentage (her mother was Christian, her father,
Jewish). Artin was not himself Jewish, but, on account of his wife's racial status in Nazi Germany, was dismissed
from his university position in 1937. They had three children, one of whom is Michael Artin, an American algebraist
currently at MIT.

Selected bibliography

• [3] Emil Artin, The theory of braids, Annals of Mathematics (2) 48 (1947), 101 - 126

• Emil Artin (1998). Galois Theory. Dover Publications, Inc.. ISBN 0-486-62342-4. (Reprinting of second revised

edition of 1944, The University of Notre Dame Press).

• A Freshman Honors Course in Calculus and Analytic Geometry ISBN 0-923891-52-8

• Emil Artin (1957), Geometric Algebra, Interscience Publishers)

• Emil Artin (1898—1962) Beitrage zu Leben, Werk und Personlichkeit, eds., Karin Reich and Alexander Kreuzer
(Dr. Erwin Rauner Verlag, Augsburg, 2007).

Emil Artin 466

Further reading

• Schoeneberg, Bruno (1970). "Artin, Emil". Dictionary of Scientific Biography. 1. New York: Charles Scribner's
Sons. pp. 306-308. ISBN 0684101 149.

References

[1] http://hup.sub.uni-hamburg.de/opus/volltexte/2008/57/pdf/HamburgUP_HUR09_Artin.pdf

[2] http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=TPRBAU000047000002000189000001&idtype=cvips&gifs=yes

[3] http://links.jstor.org/sici?sici=0003-486X%28194701%292%3A48%3Al%3C101%3ATOB%3E2.0.CO%3B2-A

[4] http://projecteuclid.Org/euclid.ndml/l 175 197041

External links

• O'Connor, John J.; Robertson, Edmund F., "Emil Artin" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/ Artin. html), MacTutor History of Mathematics archive, University of St Andrews.

• Emil Artin (http://www. genealogy. ams.org/id.php?id=7690) at the Mathematics Genealogy Project

• (http://www.princeton.edu/~mudd/finding_aids/mathoral/pmcxrota.htm) "Fine Hall in its golden age:
Remembrances of Princeton in the early fifties", by Gian-Carlo Rota. Contains a section on Artin at Princeton.

Ronald Brown 467

Ronald Brown

Born 4 January 1935London

Nationality S|S United Kingdom

Fields Mathematics

Alma mater University of Oxford

Doctoral advisor J. H. C. Whitehead

Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor
University, he has authored many books and journal articles.

Education and career

Born on 4 January 1935 in London, Brown attended Oxford University, obtaining a B.A. in 1956 and a Ph.D. in
1962. Brown began his teaching career during his Ph.D. work, serving as an assistant lecturer at Liverpool
University before assuming the position as lecturer. In 1964, he took a position at Hull University, serving first as a
senior lecturer and then as a reader before becoming a professor of pure mathematics at Bangor University, then a
part of the University of Wales, in 1970.

Brown served as professor of pure mathematics for 29 years, also serving briefly during the 1983-1984 term as an
associate professor at Louis Pasteur University in Strasbourg. In 1999, Brown took a half-time research
professorship until he became Professor Emeritus in 2001.

Editing and writing

Brown has served as an editor or on the editorial board for a number of print and electronic journals. He began in
1968 with the Chapman & Hall Mathematics Series, contributing through 1986. In 1975, he joined the editorial
advisory board of the London Mathematical Society, remaining through 1994. Two years later, he joined the
editorial board of Applied Categorical Structures, continuing through 2007. From 1995 and 1999, respectively, he
has been active with the electronic journals [Theory and Applications of Categories http://www.tac.mta.ca/tac/
] and [Homology, Homotopy and Applications http://www.intlpress.eom/7, which he helped found. Since
2006, he has been involved with Journal of Homotopy and Related Structures} His mathematical research

interests range from algebraic topology and groupoids, to homology theory, category theory, mathematical biology,
mathematical physics and higher dimensional algebra.

Brown has authored or edited a number of books and over 150 academic papers published in academic journals or
collections. His first published paper was "Ten topologies for $X\times Y$", which was published in the Quarterly
Journal of Math in 1963 Since then, his publications have appeared in many journals, including but not limited to
the Journal of Algebra, Proceedings of the American Mathematical Society, Mathematische Zeitschrift, College
Mathematics Journal, and American Mathematical Monthly. He is also known for several recent co-authored papers
on Categorical ontology.

Among his several books and standard topology and algebraic topology textbooks are: Elements of Modern Topology
(1968), Low-Dimensional Topology (1979, co-edited with T.L. Thickstun), Topology: a geometric account of
general topology, homotopy types, and the fundamental groupoid (1998), Topology and Groupoids (2006)

and Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids (EMS,
2010) [18] [19] [20] [17] [21] [22] [23] [24] [25] [26]

Ronald Brown 468

His recent fundamental results that extend the classical Van Kampen theorem to higher homotopy in higher

dimensions (HHSvKT) are of substantial interest for solving several problems in algebraic topology, both old and

[271
new. Moreover, developments in algebraic topology have often had wider implications, as for example in

algebraic geometry and also in algebraic number theory. Such higher dimensional (HHSvKT) theorems are about

homotopy invariants of structured spaces, and especially those for filtered spaces or n-cubes of spaces. A nice

example is the fact that the relative Hurewicz theorem is a consequence of HHSvKT, and this then suggested a

triadic Hurewicz theorem.

References

[I] http://www.bangor.ac.uk/~mas010/Page at Bangor University, UK

[2] "Ronald Brown: Biographical Sketch" (http://www.bangor.ac.uk/~mas010/biog-br.html). .

[3] "Applied Categorical Structures" (http://www.bangor.ac.uk/~mas010/publicfull.htm). 2010-12-08. . Retrieved 2010-12-1 1.

[4] EDITORIAL BOARD of the International journal Theory and Applications of Categories http://www.tac. mta.ca/tac/geninfo.html#edlist

[5] EDITORIAL BOARD of the International journal Homology, Homotopy and Applications http://www.intlpress.com/HHA/editors.htm

[6] http://arxiv.org/

[7] Listed in the Editors' List of the International Journal of Homotopy and Related Structures http://tcms.org.ge/Journals/JHRS/editors.htm

[8] Editor Ronald Brown's research interests: Category theory, higher dimensional algebra, holonomy, groupoids and crossed objects in algebraic

topology. http://emis.kaist.ac.kr/journals/JHRS/interests.htm
[9] Cited by John C. Baez, James Dolan., in Higher-Dimensional Algebra III: n-categories and the Algebra of Opetopes., quantum algebra and

Topology, Adv. Math. 135 (1998), 145-206.
[10] Cited by Georgescu, George and Popescu, Andrei., in "A common generalization for MV-algebras and Lukasiewicz-Moisil algebras",

Archive for Mathematical Logic, Vol. 45, No. 8. (November 2006), pp. 947-981. (in reference to Heyting-algebra higher-dimensional-algebra

hyperalgebras Lukasiewicz- Moisil-algebras meta-logics MV-algebras on 2007-07-11)

[II] Cited by John C. Baez, Laurel Langford., in "Higher-Dimensional Algebra IV: 2-Tangles." , (Quantum Algebra (math.QA); Algebraic
Topology (math. AT); Category Theory (math.CT)), Adv. Math. 180 (2003), 705-764 http://www.azimuthproject.org/azimuth/show/
John+Baez

[12] Cited in "" Higher-dimensional Algebra and Topological Quantum Field Theory" J.Math.Phys. 36 (1995) 6073-6105, by John C. Baez,

James Dolan, (2004) http://arxiv.org/abs/q-alg/9503002 doi:10.1063/1.531236
[13] "Ronald Brown Publications" (http://web.archive.Org/web/20080424002106/http://www.bangor.ac.uk/~mas010/publicfull.htm).

2008-04-19. Archived from the original (http://www.bangor.ac.uk/~mas010/publicfull.htm) on 2008-04-24. . Retrieved 2010-04-23.
[14] Cited in Online research in philosophy Entries: http://philpapers.org/
[15] Cited in Encyclopaedia of Mathematics - ISBN 1402006098
[16] Referenced in "Algebraic homotopy" http://eom.springer.de/ A/al30170.htm
[17] Cited in "Bibliography For Groupoids And Algebraic Topology" http://myyn.org/rn/article/

bibliography-for-groupoids-and-algebraic-topology/
[18] http://sz0009.ev.mail. comcast. net/service/home/~/Tracts_voll5.pdf?auth=co&loc=en_US&id=128480&part=2 and

www.ems-ph.org : EMS Tracts in Mathematics, Vol. 15
[19] http://citeseerx.ist.psu. edu/viewdoc/summary?doi=10. 1.1. 150.6444 CiteSeerX accessed on 12/10/2010
[20] http://ncatlab.org/nlab/show/Nonabelian-l- Algebraic+Topology nLab Review of Nonabelian Algebraic Topology
[21] A Review of ""Nonabelian Algebraic Topology" by Prof. J. Baez on June 6, 2009 http://golem.ph.utexas.edu/category/2009/06/

nonabelian_algebraic_topology.html
[22] Addebook • Apr 22nd, 2009 • Category: Mathematics : Nonabelian Algebraic Topology http://www.addebook.com/tech2/mathematics/

nonabelian-algebraic-topology_4164.html
[23] Cited on p. xi as a basic reference in ""Non-abelian Theories" http://myyn.org/rn/article/non-abelian-theories/
[24] [ALGTOP-L] available full draft of book on "Nonabelian algebraic topology' https://lists.lehigh.edu/pipermail/algtop-l/2009q2/

000443.html
[25] Cited in "Towards Higher Categories" By John C. Baez and J. Peter May, Publisher: Springer Verlag, Published Date: 2009-10-01, ISBN

9781441915238 http://www.isbnlib.com/author/John_C_Baez
[26] Referenced in "Nonabelian cohomology" in nLab, Revised on November 9, 2010 00:20:06 by Urs Schreiber http://ncatlab.org/nlab/

show/nonabelian+cohomology
[27] The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres and complex spaces http://mathoverflow.

net/questions/39818/the-higher-van-kampen-theorems-and-computation-of-the-unstable-homotopy-groups-of

Ronald Brown 469

External links

Ronald Brown's Biography and publications (http://planetphysics.org/encyclopedia/RonaldBrown.html)

Ronald Brown's Home Page (http://www.bangor.ac.uk/~mas010/)

Higher-Dimensional Algebra citations list (http://www.citeulike.org/tag/higher-dimensional-algebra)

Mathematical Genealogy Project page (http://genealogy.math.ndsu.nodak.edu/id.php?id=45042)

MathOverflow— A place for mathematicians to ask and answer questions, (http://mathoverflow.net/)

Editorial Board of Journal of Homotopy and Related Structures (JHRS) (http://tcms.org.ge/Journals/JHRS/

editors.htm)

nLab Abstract Mathematics Website (http://ncatlab.org/nlab/)

Editorial Board of Homology, Homotopy and Applications (HHA) (http://www.intlpress.com/HHA/)

The Origins of "Pursuing Stacks' by Alexander Grothendieck (http://www.bangor.ac.uk/~mas010/pstacks.

htm)

Journal of Homotopy and Related Structur(JHRS) Publications (http://arxiv.org/)

Homology, Homotopy and Applications (http://www.intlpress.com/)

Theory and Applications of Categories (http://www.tac.mta.ca/tac/)

Henri Poincare

470

Henri Poincare

Jules Henri Poincare (1854-
Born

1912). Photograph from the frontispiece of the 1913 edition of Last Thoughts.
29 April 1854Nancy, Meurthe-et-Moselle

Died
Residence

17 July 1912 (aged 58)Paris
France

Nationality
Fields

French

Mathematician and physicist

Institutions

Alma mater

Corps des Mines
Caen University
La Sorbonne
Bureau des Longitudes

Lycee Nancy
Ecole Polytechnique
Ecole des Mines

Doctoral advisor
Doctoral students

Charles Hermite

Louis Bachelier
Dimitrie Pompeiu
Mihailo Petrovic

Other notable students
Known for

Tobias Dantzig

Poincare conjecture

Three-body problem

Topology

Special relativity

Poincare— Hopf theorem

Poincare duality

Poincare— Birkhoff— Witt theorem

Poincare inequality

Hilbert— Poincare series

Poincare metric

Rotation number

Coining term 'Betti number'

Chaos theory

Sphere-world

Poincare— Bendixson theorem

Poincare— Lindstedt method

Poincare recurrence theorem

Influences

Lazarus Fuchs

Henri Poincare

471

Influenced

Louis Rougier
George David Birkhoff

Notable awards

RAS Gold Medal (1900)
Sylvester Medal (1901)
Matteucci Medal (1905)
Bolyai Prize (1905)
Bruce Medal (1911)

Signature

Notes

He was a cousin of Pierre Boutroux.

Jules Henri Poincare (29 April 1854 — 17 July 1912) (French pronunciation: ['3yl d'tfi pweka'Ke]) was a French
mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and
in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his
lifetime.

As a mathematician and physicist, he made many original fundamental contributions to pure and applied
mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincare
conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincare
became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos
theory. He is also considered to be one of the founders of the field of topology.

Poincare introduced the modern principle of relativity and was the first to present the Lorentz transformations in
their modern symmetrical form. Poincare discovered the remaining relativistic velocity transformations and recorded
them in a letter to Dutch physicist Hendrik Lorentz (1853—1928) in 1905. Thus he obtained perfect invariance of all
of Maxwell's equations, an important step in the formulation of the theory of special relativity.

The Poincare group used in physics and mathematics was named after him.

Life

Poincare was born on 29 April 1854 in Cite Ducale neighborhood, Nancy, Meurthe-et-Moselle into an influential
family (Belliver, 1956). His father Leon Poincare (1828—1892) was a professor of medicine at the University of
Nancy (Sagaret, 1911). His adored younger sister Aline married the spiritual philosopher Emile Boutroux. Another
notable member of Jules' family was his cousin, Raymond Poincare, who would become the President of France,

1913 to 1920, and a fellow member of the Academie francaise

[2]

Education

During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother,
Eugenie Launois (1830-1897).

In 1862, Henri entered the Lycee in Nancy (now renamed the Lycee Henri Poincare in his honour, along with the
University of Nancy). He spent eleven years at the Lycee and during this time he proved to be one of the top students
in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster
of mathematics" and he won first prizes in the concours general, a competition between the top pupils from all the
Lycees across France. His poorest subjects were music and physical education, where he was described as "average
at best" (O'Connor et al., 2002). However, poor eyesight and a tendency towards absentmindedness may explain
these difficulties (Carl, 1968). He graduated from the Lycee in 1871 with a Bachelor's degree in letters and sciences.

Henri Poincare

472

During the Franco-Prussian War of 1870 he served alongside his father in the Ambulance Corps.

Poincare entered the Ecole Polytechnique in 1873. There he studied mathematics as a student of Charles Hermite,
continuing to excel and publishing his first paper (Demonstration nouvelle des proprietes de I'indicatrice d'une
surface) in 1874. He graduated in 1875 or 1876. He went on to study at the Ecole des Mines, continuing to study
mathematics in addition to the mining engineering syllabus and received the degree of ordinary engineer in March
1879.

As a graduate of the Ecole des Mines he joined the Corps des Mines as an inspector for the Vesoul region in
northeast France. He was on the scene of a mining disaster at Magny in August 1879 in which 18 miners died. He
carried out the official investigation into the accident in a characteristically thorough and humane way.

At the same time, Poincare was preparing for his doctorate in sciences in mathematics under the supervision of
Charles Hermite. His doctoral thesis was in the field of differential equations. It was named Sur les proprietes des
fonctions definies par les equations differences. Poincare devised a new way of studying the properties of these
equations. He not only faced the question of determining the integral of such equations, but also was the first person
to study their general geometric properties. He realised that they could be used to model the behaviour of multiple
bodies in free motion within the solar system. Poincare graduated from the University of Paris in 1879.

Career

Soon after, he was offered a post as junior lecturer in mathematics
at Caen University, but he never fully abandoned his mining career
to mathematics. He worked at the Ministry of Public Services as
an engineer in charge of northern railway development from 1881
to 1885. He eventually became chief engineer of the Corps de
Mines in 1893 and inspector general in 1910.

Beginning in 1881 and for the rest of his career, he taught at the
University of Paris (the Sorbonne). He was initially appointed as
the maitre de conferences d'analyse (associate professor of
analysis) (Sageret, 1911). Eventually, he held the chairs of
Physical and Experimental Mechanics, Mathematical Physics and
Theory of Probability, and Celestial Mechanics and Astronomy.

Also in that same year, Poincare married Miss Poulain dAndecy.
Together they had four children: Jeanne (born 1887), Yvonne
(born 1889), Henriette (born 1891), and Leon (born 1893).

In 1887, at the young age of 32, Poincare was elected to the
French Academy of Sciences. He became its president in 1906,
and was elected to the Academie francaise in 1909.

In 1887 he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem
concerning the free motion of multiple orbiting bodies. (See #The three-body problem section below)

The young Henri Poincare

Henri Poincare

473

In 1893, Poincare joined the French Bureau des Longitudes, which engaged
him in the synchronisation of time around the world. In 1897 Poincare
backed an unsuccessful proposal for the decimalisation of circular measure,
and hence time and longitude (see Galison 2003). It was this post which led
him to consider the question of establishing international time zones and
the synchronisation of time between bodies in relative motion. (See #Work
on relativity section below)

In 1899, and again more successfully in 1904, he intervened in the trials of
Alfred Dreyfus. He attacked the spurious scientific claims of some of the
evidence brought against Dreyfus, who was a Jewish officer in the French
army charged with treason by anti-Semitic colleagues.

In 1912, Poincare underwent surgery for a prostate problem and
subsequently died from an embolism on 17 July 1912, in Paris. He was 58
years of age. He is buried in the Poincare family vault in the Cemetery of
Montparnasse, Paris.

A former French Minister of Education, Claude Allegre, has recently
(2004) proposed that Poincare be reburied in the Pantheon in Paris, which
is reserved for French citizens only of the highest honour.

The Poincare family grave at the Cimetiere
du Montparnasse

Students

Poincare had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu

(1905). [4]

Work

Summary

Poincare made many contributions to different fields of pure and applied mathematics such as: celestial mechanics,
fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum
theory, theory of relativity and physical cosmology.

He was also a populariser of mathematics and physics and wrote several books for the lay public.

Among the specific topics he contributed to are the following:

algebraic topology

the theory of analytic functions of several complex variables

the theory of abelian functions

algebraic geometry

Poincare was responsible for formulating one of the most famous problems in mathematics, the Poincare

conjecture.

Poincare recurrence theorem

hyperbolic geometry

number theory

the three-body problem

the theory of diophantine equations

the theory of electromagnetism

the special theory of relativity

In an 1894 paper, he introduced the concept of the fundamental group.

Henri Poincare 474

• In the field of differential equations Poincare has given many results that are critical for the qualitative theory of
differential equations, for example the Poincare sphere and the Poincare map.

• Poincare on "everybody's belief" in the Normal Law of Errors (see normal distribution for an account of that
"law")

• Published an influential paper providing a novel mathematical argument in support of quantum mechanics.

The three-body problem

The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had
eluded mathematicians since Newton's time. This was known originally as the three-body problem and later the
«-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very
important and challenging at the close of the nineteenth century. Indeed in 1887, in honour of his 60th birthday,
Oscar II, King of Sweden, advised by Gosta Mittag-Leffler, established a prize for anyone who could find the
solution to the problem. The announcement was quite specific:

Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever
collide, try to find a representation of the cooi
whose values the series converges uniformly.

collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of

In case the problem could not be solved, any other important contribution to classical mechanics would then be
considered to be prizeworthy. The prize was finally awarded to Poincare, even though he did not solve the original
problem. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as
furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its
publication will inaugurate a new era in the history of celestial mechanics. " (The first version of his contribution
even contained a serious error; for details see the article by Diacu ). The version finally printed contained many
important ideas which lead to the theory of chaos. The problem as stated originally was finally solved by Karl F.
Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s.

Work on relativity

Marie Curie and Poincare talk at the 191 1 Solvay
Conference.

Local time

Poincare's work at the Bureau des Longitudes on establishing international time zones led him to consider how
clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the
"luminiferous aether"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing
Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with
radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time"
■ff = i — vx/(? an d introduced the hypothesis of length contraction to explain the failure of optical and

191
electrical experiments to detect motion relative to the aether (see Michelson— Morley experiment). Poincare was a

Henri Poincare 475

constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincare as a philosopher was interested in
the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are
now associated with special relativity. In The Measure of Time (1898), Poincare said, " A little reflection is
sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the
result of a convention." He also argued that scientists have to set the constancy of the speed of light as a postulate to
give physical theories the simplest form. Based on these assumptions he discussed in 1900 Lorentz's "wonderful
invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals
assumed to travel with the same speed in both directions in a moving frame.

Principle of relativity and Lorentz transformations

He discussed the "principle of relative motion" in two papers in 1900 and named it the principle of relativity

in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state

ri3i

of rest. In 1905 Poincare wrote to Lorentz about Lorentz's paper of 1904, which Poincare described as a "paper of
supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation
to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by

ri4i

Lorentz. In a second letter to Lorentz, Poincare gave his own reason why Lorentz's time dilation factor was

indeed correct after all: it was necessary to make the Lorentz transformation form a group and gave what is now
known as the relativistic velocity-addition law. Poincare later delivered a paper at the meeting of the Academy of
Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote:

The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which
I will call by the name of Lorentz) of the form:

x' = U{x + et), t' = k£(t + ex), y' = ly, z' = lz, k= 1/Vl -e 2 .

and showed that the arbitrary function ^(ejmust be unity for all e (Lorentz had set £ = Iby a different
argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906
Poincare pointed out that the combination x 2 + y 2 -\- z 2 — c 2 £ 2 is invariant. He noted that a Lorentz

transformation is merely a rotation in four-dimensional space about the origin by introducing ct\/ — l as a fourth

1171 1

imaginary coordinate, and he used an early form of four-vectors. Poincare s attempt at a four-dimensional

reformulation of the new mechanics was rejected by himself in 1907, because in his opinion the translation of

11 81

physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was
Hermann Minkowski who worked out the consequences of this notion in 1907.

Mass-energy relation

Like others before, Poincare (1900) discovered a relation between mass and electromagnetic energy. While studying
the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center
of gravity still moves with a uniform velocity when electromagnetic fields are included. He noticed that the
action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum.
Poincare concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious fluid

("fluide fictif") with a mass density of Elc . If the center of mass frame is defined by both the mass of matter and the
mass of the fictitious fluid, and if the fictitious fluid is indestructible — it's neither created or destroyed — then the
motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms
of energy. So Poincare assumed that there exists a non-electric energy fluid at each point of space, into which
electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the
motion of the center of mass remains uniform. Poincare said that one should not be too surprised by these
assumptions, since they are only mathematical fictions.

However, Poincare's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain
direction, it will suffer a recoil from the inertia of the fictitious fluid. Poincare performed a Lorentz boost (to order

Henri Poincare 476

vie) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of
conservation of momentum is violated. This would allow perpetual motion, a notion which he abhorred. The laws of
nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore he
argued that also in this case there has to be another compensating mechanism in the ether.

ri3i

Poincare himself came back to this topic in his St. Louis lecture (1904). This time (and later also in 1908) he

ri9i

rejected the possibility that energy carries mass and criticized the ether solution to compensate the above
mentioned problems:

The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present
projectile has no mass; it is not matter, it is energy. [..] Shall we say that the space which separates the oscillator from the receiver and which
the disturbance must traverse in passing from one to the other, is not empty, but is filled not only with ether, but with air, or even in
inter-planetary space with some subtile, yet ponderable fluid; that this matter receives the shock, as does the receiver, at the moment the
energy reaches it, and recoils, when the disturbance leaves it? That would save Newton's principle, but it is not true. If the energy during its
propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at
least for the air, that there is nothing of the kind. Michelson and Morley have since confirmed this. We might also suppose that the motions of
matter proper were exactly compensated by those of the ether; but that would lead us to the same considerations as those made a moment ago.
The principle, if thus interpreted, could explain anything, since whatever the visible motions we could imagine hypothetical motions to
compensate them. But if it can explain anything, it will allow us to foretell nothing; it will not allow us to choose between the various possible
hypotheses, since it explains everything in advance. It therefore becomes useless.

He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass
7 m , Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2)
the non-conservation of energy in the radium experiments of Madame Curie.

It was Albert Einstein's concept of mass— energy equivalence (1905) that a body losing energy as radiation or heat

2 r201

was losing mass of amount m = Elc that resolved Poincare's paradox, without using any compensating

r2ii
mechanism within the ether. The Hertzian oscillator loses mass in the emission process, and momentum is

conserved in any frame. However, concerning Poincare's solution of the Center of Gravity problem, Einstein noted

1221

that Poincare's formulation and his own from 1906 were mathematically equivalent.

Poincare and Einstein

Einstein's first paper on relativity was published three months after Poincare's short paper, but before Poincare's

1171

longer version. It relied on the principle of relativity to derive the Lorentz transformations and used a similar
clock synchronisation procedure (Einstein synchronisation) that Poincare (1900) had described, but was remarkable
in that it contained no references at all. Poincare never acknowledged Einstein's work on special relativity. Einstein
acknowledged Poincare in the text of a lecture in 1921 called Geometrie und Erfahrung in connection with
non-Euclidean geometry, but not in connection with special relativity. A few years before his death Einstein
commented on Poincare as being one of the pioneers of relativity, saying "Lorentz had already recognised that the
transformation named after him is essential for the analysis of Maxwell's equations, and Poincare deepened this
insight still further ....

Assessments

Poincare's work in the development of special relativity is well recognised, though most historians stress that
despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the

1241

work. Poincare developed a similar physical interpretation of local time and noticed the connection to signal
velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks in the
ether show the "true" time, and moving clocks show the local time. So Poincare tried to keep the relativity principle
in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the
new physical concepts of the relativity of space and time. While this is the view of most

historians, a minority go much further, such as E.T. Whittaker, who held that Poincare and Lorentz were the true
discoverers of Relativity.

Henri Poincare

477

Character

Poincare's work habits have been compared to a bee flying from flower
to flower. Poincare was interested in the way his mind worked; he
studied his habits and gave a talk about his observations in 1908 at the
Institute of General Psychology in Paris. He linked his way of thinking
to how he made several discoveries.

The mathematician Darboux claimed he was un intuitif (intuitive),
arguing that this is demonstrated by the fact that he worked so often by
visual representation. He did not care about being rigorous and disliked
logic. He believed that logic was not a way to invent but a way to
structure ideas and that logic limits ideas.

Toulouse' characterisation

Poincare's mental organisation was not only interesting to Poincare
himself but also to Toulouse, a psychologist of the Psychology
Laboratory of the School of Higher Studies in Paris. Toulouse wrote a

[31]

■

WF*

ID "",*' ^mt

In it, he discussed Poincare's

Photographic portrait of H. Poincare by Henri
Manuel

book entitled Henri Poincare (1910).
regular schedule:

• He worked during the same times each day in short periods of time.

He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7
p.m.. He would read articles in journals later in the evening.

• His normal work habit was to solve a problem completely in his head, then commit the completed problem to
paper.

• He was ambidextrous and nearsighted.

• His ability to visualise what he heard proved particularly useful when he attended lectures, since his eyesight was
so poor that he could not see properly what the lecturer wrote on the blackboard.

These abilities were offset to some extent by his shortcomings:

• He was physically clumsy and artistically inept.

• He was always in a rush and disliked going back for changes or corrections.

• He never spent a long time on a problem since he believed that the subconscious would continue working on the
problem while he consciously worked on another problem.

In addition, Toulouse stated that most mathematicians worked from principles already established while Poincare
started from basic principles each time (O'Connor et al., 2002).

His method of thinking is well summarised as:

"Habitue' a negliger les details et a ne regarder que les cimes, il passait de Vune a I'autre avec une
promptitude surprenante et les fait s qu'il decouvrait se groupant d'eux-memes autour de leur centre etaient
instantanement et automatiquement classes dans sa memoire. "("Accustomed to neglecting details and to
looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he
discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.")
Belliver (1956)

Henri Poincare 478

Attitude towards transfinite numbers

Poincare was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which

[32]
mathematics would eventually be cured. Poincare said, "There is no actual infinite; the Cantorians have forgotten

T331
this, and that is why they have fallen into contradiction."

View on economics

Poincare saw mathematical work in economics and finance as peripheral. In 1900 Poincare commented on Louis

Bachelier's thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind

[but] the subject is somewhat remote from those our other candidates are in the habit of treating." (Bernstein, 1996,

pp. 199—200) Bachelier's work explained what was then the French government's pricing options on French Bonds

[341
and anticipated many of the pricing theories in financial markets used today.

Honours

Awards

Oscar II, King of Sweden's mathematical competition (1887)
American Philosophical Society 1899

Gold Medal of the Royal Astronomical Society of London (1900)
Bolyai Prize in 1905
Matteucci Medal 1905
French Academy of Sciences 1906
Academie Francaise 1909
Bruce Medal (1911)
Named after him

• Poincare Prize (Mathematical Physics International Prize)

• Annales Henri Poincare (Scientific Journal)

• Poincare Seminar (nicknamed "Bourbaphy")

• The crater Poincare on the Moon

• Asteroid 202 1 Poincare

Philosophy

Poincare had philosophical views opposite to those of Bertrand Russell and Gottlob Frege, who believed that
mathematics was a branch of logic. Poincare strongly disagreed, claiming that intuition was the life of mathematics.
Poincare gives an interesting point of view in his book Science and Hypothesis:

For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible,
and if the scientists are sometimes mistaken, this is only from their mistaking its rule.

Poincare believed that arithmetic is a synthetic science. He argued that Peano's axioms cannot be proven
non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic
and not analytic. Poincare then went on to say that mathematics cannot be deduced from logic since it is not analytic.
His views were similar to those of Immanuel Kant (Kolak, 2001, Folina 1992). He strongly opposed Cantorian set
theory, objecting to its use of impredicative definitions.

However, Poincare did not share Kantian views in all branches of philosophy and mathematics. For example, in
geometry, Poincare believed that the structure of non-Euclidean space can be known analytically. Poincare held that
convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be
known as "conventionalism". Poincare believed that Newton's first law was not empirical but is a conventional

Henri Poincare 479

framework assumption for mechanics. He also believed that the geometry of physical space is conventional. He

considered examples in which either the geometry of the physical fields or gradients of temperature can be changed,

either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are

expanded or shrunk by a variable heat distribution. However, Poincare thought that we were so accustomed to

Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to

[351
a non-Euclidean physical geometry.

Free will

Poincare's famous lectures before the Societe de Psychologie in Paris (published as Science and Hypothesis, The
Value of Science, and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity
and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed
by a critical evaluation.

Although he most often spoke of a deterministic universe, Poincare said that the subconscious generation of new
possibilities involves chance.

"It is certain that the combinations which present themselves to the mind in a kind of sudden
illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful
combinations... all the combinations are formed as a result of the automatic action of the subliminal ego,
but those only which are interesting find their way into the field of consciousness... A few only are
harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the
geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention
upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on
the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of

1371

discipline and to disorder born of chance."

Poincare's two stages — random combinations followed by selection — became the basis for Daniel Dennett's
two-stage model of free will.

Poincare Model of the Subconscious Mind in Mathematics

Poincare proposes a model of the Subconscious Mind stresses the Subconscious or Unconscious mind is capable of
evaluating and processing even complex mathematical or scientific ideas, and evaluate and elevate them on the basis
of their elegance and beauty

Poincare thus shares with Freud a belief that mental functions are at work in creating thinking which are not present
to our conscious mind. That the discontinuity in time between thinking of a problem and suddenly relieving a
solution proves that some mental function outside the space of conscious awareness must be at work

References

This article incorporates material from Jules Henri Poincare on PlanetMath, which is licensed under the Creative
Commons Attribution/Share-Alike License.

Footnotes and primary sources

[2] The Internet Encyclopedia of Philosophy (httpV/www.utm.edu/research/iep/p/poincare.htm) Jules Henri Poincare article by Mauro

Murzi — Retrieved November 2006.
[3] Lorentz, Poincare et Einstein (http://www.lexpress.fr/idees/tribunes/dossier/allegre/dossier.asp?ida=430274)

[4] Mathematics Genealogy Project (http://www. genealogy. ams.org/id.php?id=34227) North Dakota State University. Retrieved April 2008.
[5] McCormmach, Russell (Spring, 1967), "Henri Poincare and the Quantum Theory", his 58 (1): 37-55, doi:10.1086/350182
[6] Irons, F. E. (August, 2001), "Poincare's 191 1—12 proof of quantum discontinuity interpreted as applying to atoms", American Journal of

Physics 69 (8): 879-884, doi:10.1 1 19/1. 1356056

Henri Poincare 480

[7] Diacu, F. (1996), "The solution of the n-body Problem", The Mathematical Intelligencer 18 (3): 66-70, doi: 10.1007/BF030243 13

[8] Hsu, Jong-Ping; Hsu, Leonardo (2006), A broader view of relativity: general implications ofLorentz and Poincare invariance (http://books.

google.com/books ?id=amLqckyrvUwC), 10, World Scientific, p. 37, ISBN 9-812-56651-1, , Section A5a, p 37 (http://books.google.com/

books ?id=amLqckyrvUwC&pg=PA37)
[9] Lorentz, H.A. (1895), Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Korpern, Leiden: E.J. Brill
[10] Poincare, H. (1898), "The Measure of Time", Revue de metaphysique et de morale 6: 1—13
[11] Poincare, H. (1900), "La theorie de Lorentz et le principe de reaction", Archives neerlandaises des sciences exactes et naturelles 5: 252—278.

See also the English translation (http://www.physicsinsights.org/poincare-1900.pdf)
[12] Poincare, H. (1900), " Les relations entre la physique experimentale et la physique mathematique (http://gallica.bnf.fr/ark:/12148/

bpt6kl7075r/fl 167. table)", Revue generale des sciences pures et appliquees 11:1 163—1 175. Reprinted in "Science and Hypothesis", Ch.

9-10.
[13] Poincare, Henri (1904/6), "The Principles of Mathematical Physics", The Foundations of Science (The Value of Science), New York:

Science Press, pp. 297—320
[14] Letter from Poincare to Lorentz, Mai 1905 (http://www.univ-nancy2.fr/poincare/chp/text/lorentz3.xml)
[15] Letter from Poincare to Lorentz, Mai 1905 (http://www.univ-nancy2.fr/poincare/chp/text/lorentz4.xml)
[16] Poincare, H. (1905), "On the Dynamics of the Electron", Comptes Rendus 140: 1504—1508 (Wikisource translation)
[17] Poincare, H. (1906), "On the Dynamics of the Electron", Rendiconti del Circolo matematico Rendiconti del Circolo di Palermo 21:

129-176, doi:10.1007/BF03013466 (Wikisource translation)
[18] Walter (2007), Secondary sources on relativity
[19] Miller 1981, Secondary sources on relativity
[20] Darrigol 2005, Secondary sources on relativity
[21] Einstein, A. (1905b), " 1st die Tragheit eines Korpers von dessen Energieinhalt abhangig? (http://www.physik.uni-augsburg.de/annalen/

history/papers/1905_18_639-641.pdf)", Annalen der Physik 18: 639—643. See also English translation (http://www.fourmilab.ch/etexts/

einstein/specrel/www).
[22] Einstein, A. (1906), " Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Tragheit der Energie (http://www. physik.

uni-augsburg.de/annalen/history/papers/1906_20_627-633.pdf)", Annalen derPhysiklO (8): 627-633, doi:10.1002/andp.l9063250814
[23] Darrigol 2004, Secondary sources on relativity
[24] Galison 2003 and Kragh 1999, Secondary sources on relativity
[25] Holton (1988), 196-206
[26] Hentschel (1990), 3-13
[27] Miller (1981), 216-217
[28] Darrigol (2005), 15-18
[29] Katzir (2005), 286-288

[30] Whittaker 1953, Secondary sources on relativity
[31] Toulouse, E.,1910. Henri Poincare
[32] Dauben 1979, p. 266.
[33] Van Heijenoort, Jean (1967), From Frege to Godel: a source book in mathematical logic, 1879—1931 (http://books.google.com/

?id=v4tBTBlU05sC&pg=PA190), Harvard University Press, p. 190, ISBN 0674324498, , p 190 (http://books.google.com/

books ?id=v4tBTBlU05 sC&pg=PA 1 90)
[34] Dunbar, Nicholas (2000), Inventing money, JOHN WILEY & SONS, LTD, ISBN 0-471-4981 1-4
[35] Poincare, Henri (2007), Science and Hypothesis (http://books. google. com/books?id=2QXqHaVbkgoC), Cosimo.Inc. Press, p. 50,

ISBN 9781602065055, , Extract of page 50 (http://books.google.com/books?id=2QXqHaVbkgoC&pg=PA50#v=onepage&q&f=false)
[36] Hadamard, Jacques. An Essay On The Psychology Of Invention In The Mathematical Field. Princeton Univ Press (1949)
[37] Science and Method, Chapter 3, Mathematical Discovery, 1914, pp.58

[38] Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p. 293
[39] Ayoub, Raymond George (2004), Musings of the masters: an anthology of mathematical reflections (http://books.google.com/

books?id=WfvDhPK65agC&lpg), MAA, p. 88, ISBN 9780883855492, , Extract of page 88 (http://books.google.com/

books ?id=WfvDhPK65agC&pg=PA88)
[40] Mills, Jon (2004), Psychoanalysis at the limit: epistemology, mind, and the question of science (http://books.google.com/

books?id=IHwCB44i2JoC), SUNY Press, pp. 82-85, ISBN 9780791460658, , Extract of page 82 (http://books.google.com/

books ?id=IHwCB44i2JoC&pg=PA82)

Henri Poincare 48 1

Poincare 's writings in English translation

Popular writings on the philosophy of science:

• Poincare, Henri (1902—1908), The Foundations of Science (http://www.archive.org/details/
foundationsscieOlpoingoog), New York: Science Press; This book includes the English translations of Science
and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).

• 1913. Last Essays, (http://www.archive.org/details/mathematicsandsc001861mbp), New York: Dover reprint,
1963

On algebraic topology:

• 1895. Analysis situs. The first systematic study of topology.
On celestial mechanics:

• 1892-99. New Methods of Celestial Mechanics, 3 vols. English trans., 1967. ISBN 1-56396-1 17-2.

• 1905—10. Lessons of Celestial Mechanics.

On the philosophy of mathematics:

• Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2
vols. Oxford Univ. Press. Contains the following works by Poincare:

• 1894, "On the nature of mathematical reasoning," 972—81.

• 1898, "On the foundations of geometry," 982-1011.

• 1900, "Intuition and Logic in mathematics," 1012—20.

• 1905-06, "Mathematics and Logic, I— III," 1021-70.

• 1910, "On transfinite numbers," 1071—74.

General references

• Bell, Eric Temple, 1986. Men of Mathematics (reissue edition). Touchstone Books. ISBN 0-671-62818-6.

• Belliver, Andre, 1956. Henri Poincare ou la vocation souveraine. Paris: Gallimard.

• Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk", (p. 199-200). John Wiley & Sons.

• Boyer, B. Carl, 1968. A History of Mathematics: Henri Poincare', John Wiley & Sons.

• Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870—1940. Princeton Uni. Press.

• Dauben, Joseph (1993, 2004), " Georg Cantor and the Battle for Transfinite Set Theory (http://www.
acmsonline.org/journal/2004/Dauben-Cantor.pdf)", Proceedings of the 9th ACMS Conference (Westmont
College, Santa Barbara, CA), pp. 1—22. Internet version published in Journal of the ACMS 2004.

• Folina, Janet, 1992. Poincare and the Philosophy of Mathematics. Macmillan, New York.

• Gray, Jeremy, 1986. Linear differential equations and group theory from Riemann to Poincare, Birkhauser

• Jean Mawhin (October 2005), "Henri Poincare. A Life in the Service of Science" (http://www.ams.org/notices/
200509/comm-mawhin.pdf) (PDF), Notices of the AMS 52 (9): 1036-1044

• Kolak, Daniel, 2001. Lovers of Wisdom, 2nd ed. Wads worth.

• Murzi, 1998. "Henri Poincare" (http://www.iep.utm.edu/p/poincare.htm).

• O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincare" (http://www-history.mcs.
st-andrews. ac.uk/Mathematicians/Poincare.html). University of St. Andrews, Scotland.

• Peterson, Ivars, 1995. Newton's Clock: Chaos in the Solar System (reissue edition). W H Freeman & Co. ISBN
0-7167-2724-2.

• Sageret, Jules, 1911. Henri Poincare. Paris: Mercure de France.

• Toulouse, E.,1910. Henri Poincare (http://quod.lib. umich.edu/cgi/t/text/
text-idx?c=umhistmath;idno=AAS9989. 0001. 001). — (Source biography in French) at University of Michigan
Historic Math Collection.

Henri Poincare 482

Secondary sources to work on relativity

Cuvaj, Camillo (1969), "Henri Poincare's Mathematical Contributions to Relativity and the Poincare Stresses",
American Journal of Physics 36 (12): 1102-1113, doi: 10. 11 19/1. 1974373

Darrigol, O. (1995), "Henri Poincare's criticism of Fin De Siecle electrodynamics", Studies in History and
Philosophy of Science 26 (1): 1-44, doi:10.1016/1355-2198(95)00003-C

Darrigol, O. (2000), Electrodynamics from Ampere to Einstein, Oxford: Clarendon Press, ISBN 0198505949

Darrigol, O. (2004), "The Mystery of the Einstein— Poincare Connection" (http://www.journals.uchicago.edu/
doi/full/1 0.1086/430652), Isis 95 (4): 614-626, doi: 10. 1086/430652, PMID 16011297

Darrigol, O. (2005), "The Genesis of the theory of relativity" (http://www.bourbaphy.fr/darrigol2.pdf) (PDF),
Seminaire Poincare 1: 1—22

Galison, P. (2003), Einstein's Clocks, Poincare's Maps: Empires of Time, New York: W.W. Norton,
ISBN 0393326047

Giannetto, E. (1998), "The Rise of Special Relativity: Henri Poincare's Works Before Einstein", Atti del XVIII
congresso di storia delta fisica e dell ' astronomia: 171—207

Giedymin, J. (1982), Science and Convention: Essays on Henri Poincare's Philosophy of Science and the
Conventionalist Tradition, Oxford: Pergamon Press, ISBN 0080257909

Goldberg, S. (1967), "Henri Poincare and Einstein's Theory of Relativity", American Journal of Physics 35 (10):
934-944, doi: 10. 11 19/1. 1973643

Goldberg, S. (1970), "Poincare's silence and Einstein's relativity", British journal for the history of science 5:
73-84, doi:10.1017/S0007087400010633

Holton, G (1973/1988), "Poincare and Relativity", Thematic Origins of Scientific Thought: Kepler to Einstein,
Harvard University Press, ISBN 0674877470

Katzir, S. (2005), "Poincare's Relativistic Physics: Its Origins and Nature", Phys. Perspect. 7 (3): 268—292,
doi: 10. 1007/s00016-004-0234-y

Keswani, G.H., Kilmister, C.W. (1983), " Intimations Of Relativity: Relativity Before Einstein (http://osiris.
sunderland.ac.uk/webedit/allweb/news/Philosophy_of_Science/PIRT2002/Intimations ofRelativity.doc)",
Brit. J. Phil. Sci. 34 (4): 343-354, doi:10.1093/bjps/34.4.343

Kragh, H. (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University
Press, ISBN 0691095523

Langevin, P. (1913), "L'oeuvre d'Henri Poincare: le physicien" (http://gallica.bnf.fr/ark:/12148/bpt6kl 11418/
f93.chemindefer), Revue de metaphysique et de morale 21: 703

Macrossan, M. N. (1986), "A Note on Relativity Before Einstein" (http://espace.library.uq.edu.au/view.
php?pid=UQ:9560), Brit. J. Phil. Sci. 37: 232-234

Miller, A.I. (1973), "A study of Henri Poincare's "Sur la Dynamique de l'Electron", Arch. Hist. Exact. Scis. 10
(3-5): 207-328, doi:10.1007/BF00412332

Miller, A.I. (1981), Albert Einstein 's special theory of relativity. Emergence (1905) and early interpretation
(1905-1911), Reading: Addison-Wesley, ISBN 0-201-04679-2

Miller, A.I. (1996), "Why did Poincare not formulate special relativity in 1905?", in Jean-Louis Greffe, Gerhard
Heinzmann, Kuno Lorenz, Henri Poincare : science et philosophic, Berlin, pp. 69—100

Schwartz, H. M. (1971), "Poincare's Rendiconti Paper on Relativity. Part I", American Journal of Physics 39 (7):
1287-1294, doi: 10. 11 19/1. 1976641

Schwartz, H. M. (1972), "Poincare's Rendiconti Paper on Relativity. Part II", American Journal of Physics 40 (6):
862-872, doi: 10. 11 19/1. 1986684

Henri Poincare 483

• Schwartz, H. M. (1972), "Poincare's Rendiconti Paper on Relativity. Part III", American Journal of Physics 40
(9): 1282-1287, doi: 10. 11 19/1. 1976641

• Scribner, C. (1964), "Henri Poincare and the principle of relativity", American Journal of Physics 32 (9):
672-678, doi: 10. 11 19/1. 1986815

• Walter, S. (2005), Henri Poincare and the theory of relativity (http://www.univ-nancy2.fr/DepPhilo/walter/
papers/hpeinstein2005.htm), in Renn, J., , Albert Einstein, Chief Engineer of the Universe: 100 Authors for
Einstein (Berlin: Wiley-VCH): 162-165

• Walter, S. (2007), Breaking in the 4-vectors: the four- dimensional movement in gravitation, 1905—1910 (http://
www.univ-nancy2.fr/DepPhilo/walter/), in Renn, J., , The Genesis of General Relativity (Berlin: Springer) 3:

193-252

• Zahar, E. (2001), Poincare's Philosophy: From Conventionalism to Phenomenology, Chicago: Open Court Pub
Co, ISBN081269435X

Non-mainstream

• Keswani, G.H., (1965), "Origin and Concept of Relativity, Part I", Brit. J. Phil. Sci. 15 (60): 286-306,
doi: 10. 1093/bjps/XV.60.286

• Keswani, G.H., (1965), "Origin and Concept of Relativity, Part II", Brit. J. Phil. Sci. 16 (61): 19-32,
doi: 10. 1093/bjps/XVL6 1.19

• Keswani, G.H., (1 1966), "Origin and Concept of Relativity, Part III", Brit. J. Phil. Sci. 16 (64): 273-294,
doi: 10. 1093/bjps/XVI.64.273

• Leveugle, J. (2004), La Relativite et Einstein, Planck, Hilbert — Histoire veridique de la Theorie de la Relativiten,
Pars: L'Harmattan

• Logunov, A. A. (2004), Henri Poincare and relativity theory (http://arxiv.org/abs/physics/0408077), Moscow:
Nauka, ISBN 5-02-033964-4

• Whittaker, E.T. (1953), "The Relativity Theory of Poincare and Lorentz", A History of the Theories of Aether and
Electricity: The Modern Theories 1900—1926, London: Nelson

External links

Works by Henri Poincare (http://www.gutenberg.org/author/Henri+Poincare) at Project Gutenberg

Free audio download of Poincare's Science and Hypothesis (http://librivox.org/

science-and-hypothesis-by-henri-poincare/), from LibriVox.

Internet Encyclopedia of Philosophy: " Henri Poincare (http://www.utm. edu/research/iep/p/poincare.

htm)" — by Mauro Murzi.

Henri Poincare (http://www. genealogy. ams.org/id.php?id=34227) at the Mathematics Genealogy Project

Henri Poincare on Information Philosopher (http://www.informationphilosopher.com/solutions/scientists/

poincare/)

O'Connor, John J.; Robertson, Edmund F., "Henri Poincare" (http://www-history.mcs.st-andrews.ac.uk/

Biographies/Poincare.html), MacTutor History of Mathematics archive, University of St Andrews.

A timeline of Poincare's life (http://www.univ-nancy2.fr/ACERHP/documents/kronowww.html) University

of Nancy (in French).

Bruce Medal page (http://phys-astro.sonoma.edu/brucemedalists/Poincare/index.html)

Collins, Graham P., " Henri Poincare, His Conjecture, Copacabana and Higher Dimensions, (http://www.sciam.

com/print_version.cfm?articleID=0003848D-lC61-10C7-9C6183414B7F0000)" Scientific American, 9 June

2004.

BBC In Our Time, " Discussion of the Poincare conjecture, (http://www.bbc.co.uk/radio4/history/inourtime/

inourtime.shtml)" 2 November 2006, hosted by Melvynn Bragg. See Internet Archive (http://web.archive.org/

Henri Poincare 484

web/ */http ://w w w . bbc . co . uk/radio4/history/inourtime/inourtime. shtml)

• Poincare Contemplates Copernicus (http://www.mathpages.com/home/kmath305/kmath305.htm) at
MathPages

• H igh Anxieties - The Mathematics of Chaos (http://www.youtube.com/user/
thedebtgeneration?feature=mhum#p/u/8/5pKrKdNclYs) (2008) BBC documentary directed by David Malone
looking at the influence of Poincare's discoveries on 20th Century mathematics.

Henri Cartan

485

Henri Cartan

Born

July 8, 1904Nancy, France

Died

August 13, 2008 (aged 104)Paris, France

Nationality

| France

Fields

Mathematics

Institutions

University of Paris

Alma mater

Ecole Normale Superieure

Doctoral advisor

Paul Montel

Doctoral students

Jean-Paul Benzecri
Jean-Paul Brasselet
Pierre Cartier
Jacques Deny
Adrien Douady
Roger Godement
Max Karoubi
Jean-Louis Koszul
Jean-Pierre Serre
Rene Thorn

Known for

Cartan's theorems A and B

Notable awards

Wolf Prize (1980)

Henri Paul Cartan (July 8, 1904 — August 13, 2008) was a French mathematician with substantial contributions in

algebraic topology. He was the son of the French mathematician Elie Cartan.

Life

Cartan studied at the Lycee Hoche in Versailles, then at the ENS, receiving his doctorate in mathematics. He taught
at the University of Strasbourg from November 1931 until the outbreak of the Second World War, after which he
held academic positions at a number of other French universities, spending the bulk of his working life in Paris.

Cartan was known for work in algebraic topology, in particular on cohomology operations, the method of "killing
homotopy groups", and group cohomology. His seminar in Paris in the years after 1945 covered ground on several
complex variables, sheaf theory, spectral sequences and homological algebra, in a way that deeply influenced
Jean-Pierre Serre, Armand Borel, Alexander Grothendieck and Frank Adams, amongst others of the leading lights of
the younger generation. The number of his official students was small, but includes Adrien Douady, Roger
Godement, Max Karoubi, Jean-Pierre Serre and Rene Thom.

Henri Cartan 486

Cartan also was a founding member of the Bourbaki group and one of its most active participants. His book with

T21
Samuel Eilenberg Homological Algebra (1956) was an important text, treating the subject with a moderate level of

abstraction and category theory.

Cartan used his influence to help obtain the release of some dissident mathematicians, including Leonid Plyushch
and Jose Luis Massera. For his humanitarian efforts he received the Pagels Award from the New York Academy of

c • [3]

Sciences.

Cartan died on 13 August 2008 at the age of 104. His funeral took place the following Wednesday on 20 August in

Die, Drome.

Honours and awards

Cartan received numerous honours and awards including the Wolf Prize in 1980. From 1974 until his death he had
been a member of the French Academy of Sciences. He was a foreign member of the Finnish Academy of Science
and Letters, Royal Danish Academy of Sciences and Letters, Royal Society of London, Russian Academy of
Sciences, Royal Swedish Academy of Sciences, United States National Academy of Sciences, Polish Academy of
Sciences and other academies and societies.

Notes

[1] "Deces du mathematicien Henri Cartan" (http://www.lefigaro.fr/sciences/2008/08/18/

01008-20080818ARTFIG00240-deces-du-mathematicien-henri-cartan-.php), Le Figaro, 2008-08-18,
[2] Henri Cartan and Samuel Eilenberg, Homological Algebra ISBN 978-0-691-04991-5
[3] Notices of the AMS, Vol. 46(7), page 788 (http://www.ams.org/notices/199907/fea-cartan.pdf)

References

• O'Connor, John J.; Robertson, Edmund F., "Henri Cartan" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Cartan_Henri.html), MacTutor History of Mathematics archive, University of St Andrews.

• "Deces du mathematicien francais Henri Cartan" (http://afp.google.com/article/
ALeqM5hOohPHQlamlqDDEtglCMLPJwj30w), Agence France-Presse, 2008-08-18 (French)

• Chang, Kenneth (2008-08-25), "Henri Cartan, French Mathematician, Is Dead at 104" (http://www.nyt.imes.
com/2008/08/25/science/25cartan.html), The New York Times: A17, retrieved 2008-08-25

• Rehmeyer, Julie (2008-08-29), "Founder of the Secret Society of Mathematicians" (http://www.sciencenews.

org/view/generic/id/36064/title/Math_Trek Founder_of_the_Secret_Society_of_Mathematicians), Science

News

External links

• Henri Cartan (http://www.imdb.com/name/nm31 16760/) at the Internet Movie Database

• Jackson, Allyn (1999-07), "Interview with Henri Cartan" (http://www.ams.org/notices/199907/fea-cartan.
pdf), Notices of the American Mathematical Society 46 (7): 782—788

• Henri Cartan (http://www.academie-sciences.fr/membres/C/Cartan_Henri.htm) at lAcademie des Sciences
(French)

• Henri Cartan (http://www. genealogy. ams.org/id.php?id=49555) at the Mathematics Genealogy Project

• Biographical sketch and bibliography (http://smf.emath.fr/VieSociete/Rencontres/JourneeCartan/
NoticeCartan.html) by the Societe Mathematique de France on the occasion of Cartan's 100th birthday. (French)

• Cerf, Jean (2004-04), "Trois quarts de siecle avec Henri Cartan" (http://smf.emath.fr/Publications/Gazette/
2004/100/smf_gazette_100_7-8.pdf), Gazette des Mathematiciens (http://smf.emath.fr/Publications/Gazette/
2004/100/) (100): 7-8 (French)

Henri Cartan 487

• Samuel, Pierre (2004-04), "Souvenirs personnels sur H. Cartan" (http://smf.emath.fr/Publications/Gazette/
2004/100/smf_gazette_100_12-15.pdf), Gazette des Mathematiciens (http://smf.emath.fr/Publications/
Gazette/2004/100/) (100): 13-15 (French)

• "100th Birthday of Henri Cartan" (http://www.emis.de/newsletter/current/currentlO.pdf), European
Mathematical Society Newsletter (http://www.emis. de/newsletter/archive_contents.html#nl_53) (53): 20—21,
2004-09 (translations of above two articles from the SMF Gazette)

Jacques Hadamard

488

Jacques Hadamard

Born

Jacques Salomon Hadamard
December 8, 1865 Versailles, France

Died

October 17, 1963 (aged 97)Paris, France

Residence

France

Nationality

French

Fields

Mathematician

Institutions

University of Bordeaux
Sorbonne
College de France
Ecole Polytechnique
Ecole Centrale

Alma mater

Ecole Normale Superieure

Doctoral advisor

C. Emile Picard
Jules Tannery

Doctoral students

Maurice Rene Frechet
Paul Levy

Szolem Mandelbrojt
Andre Weil
Xinmou Wu

Known for

Hadamard product

Proof of prime number theorem
Hadamard matrices

Notable awards Grand Prix des Sciences Mathematiques
(1892)

Prix Poncelet (1898)
CNRS Gold medal (1956)

Jacques Salomon Hadamard (December 8, 1865 — October 17, 1963) was a French mathematician who made
major contributions in number theory, complex function theory, differential geometry and partial differential
equations.

Jacques Hadamard 489

Biography

The son of a teacher, Amedee Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard attended the
Lycee Charlemagne and Lycee Louis-le-Grand, where his father taught. In 1884 Hadamard entered the Ecole
Normale Superieure, having been placed first in the entrance examinations both there and at the Ecole
Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his
doctorate in 1892 and in the same year was awarded the Grand Prix des Sciences Mathematiques for his prize essay
on the Riemann zeta function.

In 1892 Hadamard married Louise- Anna Trenel, also of Jewish descent, with whom he had three sons and two
daughters. The following year he took up a lectureship in the University of Bordeaux, where he proved his
celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds. In
1896 he made two important contributions: he proved the prime number theorem, using complex function theory
(also proved independently by de la Vallee Poussin); and he was awarded the Bordin Prize of the French Academy
of Sciences for his work on geodesies in the differential geometry of surfaces and dynamical systems. In the same
year he was appointed Professor of Astronomy and Rational Mechanics in Bordeaux. His foundational work on
geometry and symbolic dynamics continued in 1898 with the study of geodesies on surfaces of negative curvature.
For his cumulative work, he was awarded the Prix Poncelet in 1898.

After the Dreyfus affair, which involved him personally because his wife was related to Dreyfus, Hadamard became
politically active and a staunch supporter of Jewish causes though he professed to be an atheist in his religion.

In 1897 he moved back to Paris, holding positions in the Sorbonne and the College de France, where he was
appointed Professor of Mechanics in 1909. In addition to this post, he was appointed to chairs of analysis at the
Ecole Polytechnique in 1912 and at the Ecole Centrale in 1920, succeeding Jordan and Appell. In Paris Hadamard
concentrated his interests on the problems of mathematical physics, in particular partial differential equations, the
calculus of variations and the foundations of functional analysis. He introduced the idea of well-posed problem and
the method of descent in the theory of partial differential equations, culminating in his seminal book on the subject,
based on lectures given at Yale University in 1922. He was elected to the French Academy of Sciences in 1916, in
succession to Poincare, whose complete works he helped edit. Later in his life he wrote on probability theory and
mathematical education. He was awarded the CNRS Gold medal for his lifetime achievements in 1956.

Hadamard's students included Maurice Frechet, Paul Levy, Szolem Mandelbrojt and Andre Weil.

On creativity

In his book Psychology of Invention in the Mathematical Field, Hadamard uses introspection to describe
mathematical thought processes. In sharp contrast to authors who identify language and cognition, he describes his
own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire
solution to a problem. He surveyed 100 of the leading physicists of the day (approximately 1900), asking them how
they did their work.

Hadamard described the experiences of the mathematicians/theoretical physicists Carl Friedrich Gauss, Hermann
von Helmholtz, Henri Poincare and others as viewing entire solutions with sudden spontaneousness.

Hadamard described the process as having four steps of the five-step Graham Wallas creative process model, with
the first three also having been put forth by Helmholtz: Preparation, Incubation, Illumination, and Verification.

Jacques Hadamard 490

Notes

[1] The Psychology of Invention in the Mathematical Field (http://press.princeton.edu/einstein/book/2dHadamard.pdf)

[2] Hadamard on Hermite (http://www-groups.dcs.st-and.ac.uk/~history/Extras/Hadamard_Hermite.html)

[3] Hadamard, 1954, pp. 13-16.

[4] Hadamard, 1954, p. 56.

Writings

• Hadamard, Jacques (2003) [1923], Lectures on Cauchy's problem in linear partial differential equations (http://
books. google. com/books ?id=B250-x21uqkC), Dover Phoenix editions, Dover Publications, New York,
ISBN 978-0-486-49549-1, JFM 49.0725.04, MR0051411

• Hadamard, Jacques (1945), The Psychology of Invention in the Mathematical Field (http://books.google.com/
books?id=iikWvQgOC5AC), Princeton University Press, ISBN 978-0-691-02931-3, MR0011665

• Hadamard, Jacques (1999) [1951], Non-Euclidean geometry in the theory of automorphic functions (http://
books. google. com/books ?isbn=0821820303), History of Mathematics, 17, Providence, R.I.: American
Mathematical Society, ISBN 978-0-8218-2030-8, MR1723250

• Hadamard, Jacques (2008) [1947], Lessons in geometry. I (http://books.google.com/
books?id=fLwydFiM7zMC), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4367-3,
MR2463454

• Hadamard, Jacques (1968), Frechet, M.; Levy, P.; Mandelbrojt, S. et al., eds., (Euvres de Jacques Hadamard.
Tomes I, II, III, TV (http://books. google.com/books ?id=XOTuAAAAMAAJ), Editions du Centre National de la
Recherche Scientifique, Paris, MR0230598

References

• Maz'ya, Vladimir; Shaposhnikova, T. O. (1998), Life and Work of Jacques Hadamard, American Mathematical
Society, ISBN 0-8218-0841-9.

• Maz'ya, V. G.; Shaposhnikova, T. O. (1998), Jacques Hadamard: a universal mathematician, History of
Mathematics, 14, American Mathematical Society/London Mathematical Society, ISBN 0821819232

External links

• O'Connor, John J.; Robertson, Edmund F., "Jacques Hadamard" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Hadamard.html), MacTutor History of Mathematics archive, University of St Andrews.

• Jacques Hadamard (http://www. genealogy. ams.org/id.php?id=24555) at the Mathematics Genealogy Project

Niels Bohr

491

Niels Bohr

Born
Died

Niels Henrik David Bohr7 October 1885Copenhagen, Denmark
18 November 1962 (aged 77)Copenhagen, Denmark

Nationality
Fields

Danish
Physics

Institutions

Alma mater

University of Copenhagen
University of Cambridge
University of Manchester

University of Copenhagen

Doctoral advisor
Other academic advisors

Christian Christiansen

J. J. Thomson
Ernest Rutherford

Doctoral students
Known for

Hendrik Anthony Kramers

Copenhagen interpretation
Complementarity
Bohr model

Sommerfeld— Bohr theory
BKS theory
Bohr-Einstein debates
Bohr magneton

Influences
Influenced

Ernest Rutherford

Werner Heisenberg
Wolfgang Pauli
Paul Dirac
Lise Meitner
Max Delbriick
and many others

Notable awards Nobel Prize in Physics (1922)

Franklin Medal (1929)

Signature

Notes

Harald Bohr is his younger brother, and Aage Bohr is his son.

Niels Bohr 492

Niels Henrik David Bohr (Danish pronunciation: [nels boe 7 ]; 7 October 1885 — 18 November 1962) was a Danish
physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which
he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of
the century at his institute in Copenhagen. He was part of a team of physicists working on the Manhattan Project.
Bohr married Margrethe N0rlund in 1912, and one of their sons, Aage Bohr, grew up to be an important physicist
who in 1975 also received the Nobel prize. Bohr has been described as one of the most influential scientists of the
20th century. [1]

Biography
Early years

Bohr was born in Copenhagen, Denmark, in 1885. His father, Christian Bohr, was professor of physiology at the
University of Copenhagen (it is his name which is given to the Bohr shift or Bohr effect), while his mother, Ellen
Adler Bohr, came from a wealthy Jewish family prominent in Danish banking and parliamentary circles. His brother
was Harald Bohr, a mathematician and Olympic footballer who played on the Danish national team. Niels Bohr was
a passionate footballer as well, and the two brothers played a number of matches for the Copenhagen-based
Akademisk Boldklub, with Niels in goal. There is, however, no truth in the oft-repeated claim that Niels Bohr
emulated his brother Harald by playing for the Danish national team.

In 1903 Bohr enrolled as an undergraduate at Copenhagen University, initially studying philosophy and
mathematics. In 1905, prompted by a gold medal competition sponsored by the Royal Danish Academy of Sciences
and Letters, he conducted a series of experiments to examine the properties of surface tension, using his father's
laboratory in the university, familiar to him from assisting there since childhood. His essay won the prize, and it was
this success that decided Bohr to abandon philosophy and adopt physics. As a student under Christian Christiansen
he received his doctorate in 1911. As a post-doctoral student, Bohr first conducted experiments under J. J. Thomson
at Trinity College, Cambridge. In 1912 he joined Ernest Rutherford at Manchester University and he adapted
Rutherford's nuclear structure to Max Planck's quantum theory and so obtained a theory of atomic structure which,
with later improvements, mainly as a result of Heisenberg's concepts, remains valid to this day. On the basis of
Rutherford's theories, Bohr published his model of atomic structure in 1913, introducing the theory of electrons
traveling in orbits around the atom's nucleus, the chemical properties of the element being largely determined by the
number of electrons in the outer orbits. Bohr introduced the idea that an electron could drop from a higher-energy
orbit to a lower one, emitting a photon (light quantum) of discrete energy. This became a basis for quantum theory.
After four productive years with Ernest Rutherford in Manchester, Bohr returned to Denmark becoming in 1918
director of the newly created Institute of Theoretical Physics.

Bohr and his wife Margrethe N0rlund Bohr had six sons. Their oldest died in a tragic boating accident and another
died from childhood meningitis. The others went on to lead successful lives, including Aage Bohr, who became a
very successful physicist and, like his father, won a Nobel Prize in physics, in 1975.

Physics

In 1916, Bohr became a professor at the University of Copenhagen. With the assistance of the Danish government

and the Carlsberg Foundation, he succeeded in founding the Institute of Theoretical Physics in 1921, of which he

mi
became director. In 1922, Bohr was awarded the Nobel Prize in physics "for his services in the investigation of the

structure of atoms and of the radiation emanating from them." Bohr's institute served as a focal point for theoretical

physicists in the 1920s and '30s, and most of the world's best known theoretical physicists of that period spent some

time there.

Niels Bohr

493

Niels Bohr as a young man. Exact
date of photo unknown.

Bohr also conceived the principle of complementarity: that items could be
separately analyzed as having several contradictory properties. For example,
physicists currently conclude that light behaves either as a wave or a stream of
particles depending on the experimental framework — two apparently mutually
exclusive properties — on the basis of this principle. Bohr found philosophical
applications for this daringly original principle. Albert Einstein much preferred
the determinism of classical physics over the probabilistic new quantum physics
(to which Max Planck and Einstein himself had contributed). He and Bohr had
good-natured arguments over the truth of this principle throughout their lives
(see Bohr— Einstein debates).

Werner Heisenberg worked as an assistant to Bohr and university lecturer in
Copenhagen from 1926 to 1927. It was in Copenhagen, in 1927, that Heisenberg
developed his uncertainty principle, while working on the mathematical
foundations of quantum mechanics. Heisenberg later became head of the German
atomic bomb project. In 1941, during the German occupation of Denmark in
World War II, Bohr was visited by Heisenberg in Copenhagen (see section
below). In 1943, shortly before he was to be arrested by the German police, Bohr
escaped to Sweden, and then traveled to London.

Atomic research

Bohr worked at the top-secret Los Alamos laboratory in New Mexico, U.S., on
the Manhattan Project, where he was known by the assumed name of Nicholas
Baker for security reasons. His role in the project was important as he was a
knowledgeable consultant or "father confessor" on the project. He was concerned
about a nuclear arms race, and is quoted as saying, "That is why I went to
America. They didn't need my help in making the atom bomb

.,[61

Niels Bohr and Albert Einstein

debating quantum theory at Paul

Ehrenfest's home in Leiden

(December 1925).

Bohr believed that atomic secrets should be shared by the international scientific
community. After meeting with Bohr, J. Robert Oppenheimer suggested that
Bohr visit President Franklin D. Roosevelt to convince him that the Manhattan
Project should be shared with the Russians in the hope of speeding up its results.
Roosevelt suggested that Bohr return to the United Kingdom to try to win British approval. Winston Churchill
disagreed with the idea of openness towards the Russians to the point that he wrote in a letter: "It seems to me Bohr
ought to be confined or at any rate made to see that he is very near the edge of mortal crimes.

„[7]

Niels Bohr

494

Coat of arms

After the war Bohr returned to Copenhagen, advocating the peaceful use of
nuclear energy. When awarded the Order of the Elephant by the Danish
government, he designed his own coat of arms which featured a taijitu

(symbol of yin and yang) and the Latin motto contraria sunt complementa:

rsi
opposites are complementary. He died in Copenhagen in 1962 of heart

failure. He is buried in the Assistens Kirkegard in the N0rrebro section of

Copenhagen.

Political activity

As regards the Occupation of Denmark during World War II, and especially
the events surrounding the Danish policy of cooperation with Nazi Germany
and the treatment of Danish Jews, most Danish archives remained sealed until
1998, many remain sealed to this day, and opinions of Danish historical
scholars and politicians on these dark topics remain deeply divided. So
too, scholarly opinions remain divided on the importance of the political
activities of Niels Bohr during this period. All sources agree that almost as soon as Hitler had taken power in
Germany Bohr played an active role in rescuing Jewish physicists out of Germany, typically offering them haven in
Copenhagen before they could take up permanent residence elsewhere. As for Sweden during World War II and
especially in the autumn of 1943, it was far from certain that they would accept Danish Jews attempting to escape
Hitler's deportation order. As related by Bohr's friend Stefan Rozental and the historian Richard Rhodes (sources
cited in note 1 1), Bohr was immediately smuggled out of Denmark in order to secure his services for the Manhattan
project. But rather than proceeding promptly to the United States, as had been planned for him, on 30 September
1943 Bohr persuaded King Gustav of Sweden to make public Sweden's willingness to provide asylum, on 2 October
1943 Swedish radio broadcast that Sweden was ready to offer asylum, and there followed quickly thereafter the mass
Rescue of the Danish Jews by their countrymen. Historians are divided not on Bohr's political actions in Sweden, but
rather on the implications and impacts of those actions. Some argue that Bohr was among those rescued and
therefore could have played no role in facilitating the mass rescue, whereas Richard Rhodes and others (note 11)
interpret Bohr's political action in Sweden as being a decisive event without which that mass rescue could not have
occurred. Whether or not the mass Rescue of the Danish Jews could have happened without Bohr's political activity
in Sweden, there is no doubt that he did all that he could for his countrymen.

Contributions to Physics and Chemistry

• The Bohr model of the atom, the theory that electrons travel in discrete orbits around the atom's nucleus.

• The shell model of the atom, where the chemical properties of an element are determined by the electrons in the
outermost orbit.

• The correspondence principle, the basic tool of Old quantum theory.

• The liquid drop model of the atomic nucleus.

235 ri21

• Identified the isotope of uranium that was responsible for slow-neutron fission - U.

• Much work on the Copenhagen interpretation of quantum mechanics.

• The principle of complementarity: that items could be separately analyzed as having several contradictory
properties.

Niels Bohr 495

Kierkegaard's influence on Bohr

It is generally accepted that Bohr read the 19th century Danish Christian existentialist philosopher, S0ren
Kierkegaard. Richard Rhodes argues in The Making of the Atomic Bomb that Bohr was influenced by Kierkegaard
via the philosopher Harald H0ffding, who was strongly influenced by Kierkegaard and who was an old friend of
Bohr's father. In 1909, Bohr sent his brother Kierkegaard's Stages on Life's Way as a birthday gift. In the enclosed
letter, Bohr wrote, "It is the only thing I have to send home; but I do not believe that it would be very easy to find

anything better.... I even think it is one of the most delightful things I have ever read." Bohr enjoyed Kierkegaard's

F131
language and literary style, but mentioned that he had some "disagreement with [Kierkegaard's ideas].'

Given this, there has been some dispute over whether Kierkegaard influenced Bohr's philosophy and science. David

ri4i
Favrholdt argues that Kierkegaard had minimal influence over Bohr's work, taking Bohr's statement about

disagreeing with Kierkegaard at face value; while Jan Faye endorses the opposing point of view by arguing that

one can disagree with the content of a theory while accepting its general premises and structure.

Relationship with Heisenberg

Bohr and Werner Heisenberg enjoyed a strong mentor/protege relationship up to the onset of World War II. Bohr
became aware of Heisenberg's talent during a lecture Heisenberg gave in Gottingen in 1922. During the mid-1920s,
Heisenberg worked with Bohr at the institute in Copenhagen. Heisenberg, like most of Bohr's assistants, learned
Danish. Heisenberg's uncertainty principle was developed during this period, as was Bohr's complementarity
principle.

By the time of World War II, the relationship became strained; this was in part because Bohr, with his partially
Jewish heritage, remained in occupied Denmark, while Heisenberg remained in Germany and became head of the
German nuclear effort. Heisenberg made a famous visit to Bohr in September 1941 and during a private moment it

seems that he began to address nuclear energy and morality as well as the war. Neither Bohr nor Heisenberg spoke

ri7i
about it in any detail or left written records of this part of the meeting and they were alone and outside. Bohr

seems to have reacted by terminating that conversation abruptly while not giving Heisenberg hints in any direction.

While some suggest that the relationship became strained at this meeting, other evidence shows that the level of
contact had been reduced considerably for some time already. Heisenberg suggested that the fracture occurred later.
In correspondence to his wife, Heisenberg described the final visit of the trip: "Today I was once more, with
Weizsacker, at Bohr's. In many ways this was especially nice, the conversation revolved for a large part of the
evening around purely human concerns, Bohr was reading aloud, I played a Mozart Sonata (A-Major). Ivan

Supek, one of Heisenberg's students and friends, claimed that the main figure of the meeting was actually

ri9i

Weizsacker who tried to persuade Bohr to mediate peace between Great Britain and Germany.

Tube Alloys

"Tube Alloys" was the code-name for the British nuclear weapon program. British intelligence inquired about Bohr's
availability for work or insights of particular value. Bohr's reply made it clear that he could not help. This reply, like
his reaction to Heisenberg, made sure that if Gestapo intercepted anything attributed to Bohr it would point to no
knowledge regarding nuclear energy as it stood in 1941. This does not exclude the possibility that Bohr privately
made calculations going further than his work in 1939 with Wheeler.

After leaving Denmark in the dramatic day and night (October 1943) when most Jews were able to escape to Sweden
due to exceptional circumstances (see Rescue of the Danish Jews), Bohr was quickly asked again to join the British
effort and he was flown to the UK. He was evacuated from Stockholm in 1943 in an unarmed De Havilland
Mosquito operated by British Overseas Airways Corporation (BOAC). Passengers on BOAC's Mosquitos were
carried in an improvised cabin in the bomb bay. The flight almost ended in tragedy as Bohr did not don his oxygen
equipment as instructed and passed out at high altitude. He would have died had not the pilot surmising from Bohr's

Niels Bohr 496

lack of response to intercom communication that he had lost consciousness, descended to a lower altitude for the
remainder of the flight. Bohr's comment was that he had slept like a baby for the entire flight.

As part of the UK team on "Tube Alloys" Bohr went to Los Alamos. Oppenheimer credited Bohr warmly for his
guiding help during certain discussions among scientists there. Discreetly, he met President Franklin D. Roosevelt
and later Winston Churchill to warn against the perilous perspectives that would follow from separate development
of nuclear weapons by several powers rather than some form of controlled sharing of the knowledge, which would
spread quickly in any case. Only in the 1950s, after the Soviet Union's first nuclear weapon test, was it possible to
create the International Atomic Energy Agency along the lines of Bohr's suggestion.

Speculation

In 1957, while the author Robert Jungk was working on the book Brighter Than a Thousand Suns, Heisenberg wrote
to Jungk explaining that he had visited Copenhagen to communicate to Bohr his view that scientists on either side
should help prevent development of the atomic bomb, that the German attempts were entirely focused on energy
production and that Heisenberg's circle of colleagues tried to keep it that way. Heisenberg acknowledged that his
cryptic approach of the subject had so alarmed Bohr that the discussion failed. Heisenberg nuanced his claims and
avoided the implication that he and his colleagues had sabotaged the bomb effort; this nuance was lost in Jungk's
original publication of the book, which implied that the German atomic bomb project was obstructed by Heisenberg.

When Bohr saw Jungk's erroneous depiction in the Danish translation of the book, he disagreed. He drafted (but
never sent) a letter to Heisenberg, stating that while Heisenberg had indeed discussed the subject of nuclear weapons

in Copenhagen, Heisenberg had never alluded to the fact that he might be resisting efforts to build such weapons.

T211
Bohr dismissed the idea of any pact as hindsight.

Michael Frayn's play Copenhagen, which was performed in London (for five years), Copenhagen, Gothenburg,
Rome, Athens, Geneva and on Broadway in New York, explores what might have happened at the 1941 meeting
between Heisenberg and Bohr. Frayn points in particular to the onus of being one of the few to understand what it
would mean to create a nuclear weapon.

Open World

Bohr advocated informing the Soviet authorities that the atomic bomb would soon be in use. In 1944 he obtained an

T221
audience with Winston Churchill, who became worried about whether Bohr was a security risk. In 1950 he

[231
addressed an 'Open Letter' to the United Nations.

Legacy

T241

• He was one of the founding fathers of CERN in 1954.

• Received the first ever Atoms for Peace Award in 1957.

• In 1965, three years after Bohr's death, the Institute of Physics at the University of Copenhagen changed its name
to the Niels Bohr Institute.

• The Bohr models semicentennial was commemorated in Denmark on 21 November 1963 with a postage stamp
depicting Bohr, the hydrogen atom and the formula for the difference of any two hydrogen energy levels:

hv = e 2 - e 1 .

• Bohrium (a chemical element, atomic number 107) is named in honour of Bohr.

• Hafnium, another chemical element, whose properties were predicted by Bohr, was named by him after Hafnia,
Copenhagen's Latin name.

• Asteroid 3948 Bohr is named after him.

• The Centennial of Bohr's birth was commemorated in Denmark on 3 October 1985 with a postage stamp
depicting Bohr with his wife Margrethe.

Niels Bohr 497

• In 1997 the Danish National Bank started circulating the 500-krone banknote with the portrait of Bohr smoking a

• [25] [26]
pipe.

• Bohr has been a common name in Europe since the Middle Ages. It remains fairly common in Europe and
spread to the U.S. with pilgrims named Bohr settling there. There was an notable increase in the middle name
Bohr throughout Europe and America following Bohr's death.

• Bohr was referenced in The Simpsons thirteenth season episode entitled I Am Furious Yellow. In the episode,
Homer Simpson is looking forward to watching a television show called When Dinosaurs Get Drunk, when it is
suddenly announced that it will be replaced with another called The Boring World of Niels Bohr. The opening
image is of the scientist and an expanded version of the Bohr model of the atom.

Notes

[I] Murdoch, Dugald (2000) "Bohr" in Newton-Smith, N. H. (ed.) A Companion to the Philosophy of Science. Great Britain: Blackwell
Publishers, p. 26. ISBN 0-631-23020-3.

[2] James Dart (2005-07-27). "Bohr's footballing career" (http://www.guardian.co.uk/football/2005/jul/27/theknowledge.panathinaikos).

London: Guardian. . Retrieved 2010-03-10.
[3] Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Simon and Schuster, pp. 62-63. ISBN 0-671-44133-7.
[4] Finn Aaserud. "History of the institute: The establishment of an institute" (http://web.archive.Org/web/20080405160424/http://www.

nbi.ku.dk/english/about/history/). Niels Bohr Institute. Archived from the original (http://www.nbi.ku.dk/english/about/history/) on

2008-04-05. . Retrieved 2008-05-11.
[5] Pais, Abraham. Niels Bohr's Times, In Physics, Philosophy and Polity p.496
[6] Long, Doug. "Niels Bohr - The Atomic Bomb and beyond" (http://www.doug-long.com/bohr.htm). Hiroshima - was it necessary?. .

Retrieved 2006-12-21.
[7] Rhodes (1986:528-538)

[8] "Bohr crest" (http://www.nbi.dk/hehi/logo/bohr_crest.png). University of Copenhagen. 1947-10-17. . Retrieved 2007-03-16.
[9] Pais, Abraham. Niels Bohr's Times, In Physics, Philosophy and Polity p. 529
[10] Vilhjalmsson, Vilhjalmur Orn & Blildnikow, Bent. Rescue, Expulsion, and Collaboration: Denmark's Difficulties with its World War II Past

(http://www.jcpa.org/phas/phas-vilhjalmsson-f06.htm) Jewish Political Studies Review 18:3—4 (Fall 2006).

[II] Each of these sources describe the political activity of Bohr before and during the Nazi era and his activities during the Rescue of the Danish
Jews. Each is accessible on Google books.

• "Niels Bohr: collected works. The political arena (1934-1961)", Pagel4, Niels Bohr, Leon Rosenfeld, Finn Aaserud, Elsevier, 2005.

• "The making of the atomic bomb." pages 483-484, Richard Rhodes, Simon and Schuster, 1986.

• "Darkness over Denmark: the Danish resistance and the rescue of the Jews." Ellen Levine, Holiday House, 2000.

• "The Rescue of the Danish Jews: moral courage under stress." page 10, Leo Goldberger, NYU Press, 1987.

• "Blank Spots on the Map: The Dark Geography of the Pentagon's Secret World", Trevor Paglen, Penguin, Penguin, 2010.

• "The destruction of the European Jews." Volume 2, page 596, Raul Hilberg, Yale University Press, 2003.

• "The Holocaust: A History of Courage and Resistance." page 136, Bea Stadtler, Morrison David Beal, David Stone Martin, Behrman
House, Inc, 1995.

• "Resistance fighter: a personal history of the Danish resistance." Pages 91-93, J0rgen Kieler, Gefen Publishing House Ltd, 2007

• "Niels Bohr: His life and work as seen by his friends and colleagues.", page 168, Stefan Rozental, North-Holland, 1967.

• "Heisenberg's war: the secret history of the German bomb." page 235, Thomas Powers, Da Capo Press , 2000.
[12] Rhodes (1986:282-88)

[13] Register, Bryan (1997-12-01). "Complementarity: Content, Context and Critique" (http://enlightenment.supersaturated.com/essays/text/

bryanregister/bohr_compliementarity.html). . Retrieved 2006-12-21.
[14] Favrholdt, David. Niels Bohr's Philosophical Background. Copenhagen: Munksgaard (1992): pp. 42-63.
[15] Faye, Jan. "Niels Bohr: His Heritage and Legacy." Dordrecht: Kluwer Academic Publishers (1991).
[16] Mark Richardson, et al. Religion & Science: History, Method, Dialogue. Routledge 1996, pg.289
[17] Heisenberg, Elisabeth (1984). Inner Exile: Recollections of a Life With Werner Heisenberg. Boston MA: Birkhauser. p. 77 et seq.

ISBN 0817631461.
[18] Heisenberg, Werner. "Letter from Werner Heisenberg to his wife Elisabeth written during his 1941 visit in Copenhagen" (http://

werner-heisenberg.unh.edu/copenhagen.htm). Heisenberg, Jochen. . Retrieved 2006-12-21.
[19] Jutarnji list. "A March 2006 interview with Ivan Supek relating to 1941 Bohr - Heisenberg meeting (Croatian)" (http://jutarnji.hr/clanak/

art-2006,3,19,supek_interyju,17440.jl?artpg=l). Jutarnji list. . Retrieved 2007-08-13.
[20] Heisenberg, Werner. "Letter From Werner Heisenberg to Author Robert Jungk" (http://web.archive.Org/web/20061017232033/http://

childrenofthemanhattanproject.org/MP_Misc/Bohr_Heisenberg/bohr_2.htm). The Manhattan Project Heritage Preservation Association,

Inc.. Archived from the original (http://www.childrenofthemanhattanproject.org/MP_Misc/Bohr_Heisenberg/bohr_2.htm) on

2006-10-17. . Retrieved 2006-12-21.

Niels Bohr 498

[21] Aaserud, Finn (2002-02-06). "Release of documents relating to 1941 Bohr-Heisenberg meeting" (http://www.nba.nbi.dk/release.html).

Niels Bohr Archive. . Retrieved 2007-06-04.
[22] Niels Bohr's mission for an 'open world' (http://www.2iceshs.cyfronet.pl/2ICESHS_Proceedings/Chapter_25/R-17_Aaserud.pdf).

Retrieved 15 November 2008.
[23] To the United Nations (http://www.seas.columbia.edu/~ah297/un-esa/wsl999-letter-bohr.html) by Niels Bohr accessed 15 November

2008
[24] CERN History (http://public.web.cern.ch/public/en/about/History54-en.html)
[25] The coins and banknotes of Denmark (http://www.nationalbanken.dk/C1256BE900406EF3/sysOakFil/Danmarks_penge_2005_ENG/

$File/Coins_Banknotes.pdf). Danmarks Nationalbank. 2005. pp. 20-21. ISBN 87-87251-55-8. . Retrieved 7 September 2010.
[26] "500-krone banknote, 1997 series" (http://www.nationalbanken.dk/DNUK/NotesAndCoins.nsf/side/500-kronelOpenDocument).

Danmarks Nationalbank. . Retrieved 7 September 2010.
[27] Family crest (http://www.houseofnames.com/xq/asp.fc/qx/bohr-family-crest.htm)

Further reading

Niels Bohr (1913). " On the Constitution of Atoms and Molecules, Part I (http://web.ihep.su/dbserv/compas/src/
bohrl3/eng.pdf)". Philosophical Magazine 26: 1—24. The landmark paper laying the Bohr model of the atom and
molecular bonding.

• 1999. Causality and Complementarity: Epistemological Lessons of Studies in Atomic Physics. Ox Bow Press.
ISBN 1-881987-13-2. The 1949-50 Gifford lectures.

• 1987 (1958). Atomic Physics and Human Knowledge. Ox Bow Press. ISBN 0-91802452-8. Seven essays written
1933-57. 1958 ed., Wiley Interscience.

• Niels Bohr Collected Works 13-Volume Limited Edition Set, General Editor, Finn Aaserud; ISBN
978-0-444-53286-2.

• Niels Bohr: The Man, His Science, and the World They Changed, by Ruth Moore; ISBN 0-262-63101-6.

• Niels Bohr's Times, In Physics, Philosophy and Polity, by Abraham Pais; ISBN 0-19-852049-2.

• Niels Bohr: His Life and Work As Seen by His Friends and Colleagues, edited by Stefan Rozental, John Wiley &
Sons, 1964.

• Suspended In Language: Niels Bohr's Life, Discoveries, And The Century He Shaped by Jim Ottaviani (graphic
novel); ISBN 0-9660106-5-5.

• Harmony and Unity : The Life ofNiel's Bohr, by Niels Blaedel; ISBN 0-910239-14-2.

• Niels Bohr: A Centenary Volume, edited by A. P French and P.J. Kennedy. ISBN 0-674-62415-7.

• Copenhagen Michael Frayn ISBN 0413724905

• Faust in Copenhagen: A Struggle for the Soul of Physics by Gino Segre; ISBN 0-670-03858-X.

External links

• Niels Bohr Collected Works, (http://elsevierdirect.com/nielsbohr)

• Niels Bohr on the 500 Danish Kroner banknote, (http://www-personal.umich.edu/~jbourj/moneyl.htm)

• Encyclopaedia Britannica article on Niels Bohr (http://www.britannica.com/eb/article-9106088/Niels-Bohr)

• Niels Bohr Archive (http://www.nba.nbi.dk/)

• Nobel Foundation: Niels Bohr (http://www.nobel.se/physics/laureates/1922/)

• Annotated bibliography for Niels Bohr from the Alsos Digital Library for Nuclear Issues (http://alsos.wlu.edu/
qsearch.aspx?browse=people/Bohr,+Niels)

• Quantum Chemistry I Lecture - Bohr's Model of the Atom (http://cinarz.zdo.com/moodle/mod/resource/
view.php?id=15)

• The Bohr-Heisenberg meeting in September 1941 (http://www.aip.org/history/heisenberg/
bohr-heisenberg-meeting . htm)

• Werner Heisenberg Ausstellung: Vom Frieden zum Krieg: Kernphysik und Kernenergie (http://www.archiv.
uni-leipzig.de/heisenberg/Vom_Frieden_zum Krieg/vom_frieden_zum krieg.htm)

Niels Bohr 499

• Oral History interview transcript with Niels Bohr 31 October 1962, American Institute of Physics, Niels Bohr
Library and Archives (http://www.aip.org/history/ohilist/4517_l.html)

• Short video documentary about Niels Bohr (http://www.science.tv/watch/3bcea3b7513ccef5857a/
THE-CORE)

• The gunfighter's dilemma (http://news.bbc.co.uk/today/hi/today/newsid_8493000/8493203.stm) BBC
News story about Bohr's researches on reaction times.

Werner Heisenberg

500

Werner Heisenberg

Born
Died

Werner Karl Heisenberg5 December 1901Wurzburg, Germany
1 February 1976 (aged 74)Munich, Germany

Nationality
Fields

German
Physics

Institutions

Alma mater

University of Gottingen
University of Copenhagen
University of Leipzig
University of Berlin
University of Munich

University of Munich

Doctoral advisor
Other academic advisors

Arnold Sommerfeld

Niels Bohr
Max Born

Doctoral students

Other notable students

Felix Bloch
Edward Teller
Rudolph E. Peierls
Reinhard Oehme
Friedwardt Winterberg
Peter Mittelstaedt
§erban Tijeica
Ivan Supek
Erich Bagge
Hermann Arthur Jahn
Raziuddin Siddiqui
Heimo Dolch
Hans Heinrich Euler
Edwin Gora
Bernhard Kockel
Arnold Siegert
Wang Foh-san

William Vermillion Houston
Guido Beck
Ugo Fano

Werner Heisenberg 501

Known for

Uncertainty Principle

Heisenberg's microscope

Matrix mechanics

Kramers-Heisenberg formula

Heisenberg group

Isospin

Influenced

Robert Dopel

Carl Friedrich von Weizsacker

Notable awards

Nobel Prize in Physics (1932)

Max Planck Medal (1933)

Notes

He was the father of the neurobiologist Martin Heisenberg and the son of August Heisenberg

Werner Heisenberg (5 December 1901 — 1 February 1976) was a German theoretical physicist who made
foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of
quantum theory. In addition, he made important contributions to nuclear physics, quantum field theory, and particle
physics.

Heisenberg, along with Max Born and Pascual Jordan, set forth the matrix formulation of quantum mechanics in
1925. Heisenberg was awarded the 1932 Nobel Prize in Physics for the creation of quantum mechanics, and its
application especially to the discovery of the allotropic forms of hydrogen.

Following World War II, he was appointed director of the Kaiser Wilhelm Institute for Physics, which was soon
thereafter renamed the Max Planck Institute for Physics. He was director of the institute until it was moved to
Munich in 1958, when it was expanded and renamed the Max Planck Institute for Physics and Astrophysics.

Heisenberg was also president of the German Research Council, chairman of the Commission for Atomic Physics,
chairman of the Nuclear Physics Working Group, and president of the Alexander von Humboldt Foundation.

Biography
Early years

Heisenberg was born in Wiirzburg, Germany to Kaspar Earnesta August Heisenberg, a secondary school teacher of
classical languages who became Germany's only ordentlicher Professor (ordinarius professor) of medieval and
modern Greek studies in the university system, and his wife Annie Wecklein.

He studied physics and mathematics from 1920 to 1923 at the Ludwig-Maximilians-Universitat Munchen and the
Georg-August-Universitdt Gottingen. At Munich, he studied under Arnold Sommerfeld and Wilhelm Wien. At
Gottingen, he studied physics with Max Born and James Franck, and he studied mathematics with David Hilbert. He
received his doctorate in 1923, at Munich under Sommerfeld. He completed his Habilitation in 1924, at Gottingen
under Born.

Because Sommerfeld had a sincere interest in his students and knew of Heisenberg's interest in Niels Bohr's theories
on atomic physics, Sommerfeld took Heisenberg to Gottingen to the Bohr-Festspiele (Bohr Festival) in June 1922.
At the event, Bohr was a guest lecturer and gave a series of comprehensive lectures on quantum atomic physics.
There, Heisenberg met Bohr for the first time, and it had a significant and continuing effect on him.

ro]

Heisenberg's doctoral thesis, the topic of which was suggested by Sommerfeld, was on turbulence; the thesis
discussed both the stability of laminar flow and the nature of turbulent flow. The problem of stability was

investigated by the use of the Orr— Sommerfeld equation, a fourth order linear differential equation for small

T91
disturbances from laminar flow. He briefly returned to this topic after World War II.

Heisenberg's paper on the anomalous Zeeman effect was accepted as his Habilitationsschrift under Max Born at
Gottingen.

Werner Heisenberg 502

In his youth he was a member and Scoutleader of the Neupfadfinder, a German Scout association and part of the
German Youth Movement. In August 1923 Robert Honsell and Heisenberg organized a trip (Grofifahrt) to

Finland with a Scout group of this association from Munich.

Career

Gottingen, Copenhagen, and Leipzig

From 1924 to 1927, Heisenberg was a Privatdozent at Gottingen. From 17 September 1924 to 1 May 1925, under an
International Education Board Rockefeller Foundation fellowship, Heisenberg went to do research with Niels Bohr,
director of the Institute of Theoretical Physics at the University of Copenhagen. He returned to Gottingen and with
Max Born and Pascual Jordan, over a period of about six months, developed the matrix mechanics formulation of
quantum mechanics. On 1 May 1926, Heisenberg began his appointment as a university lecturer and assistant to
Bohr in Copenhagen. It was in Copenhagen, in 1927, that Heisenberg developed his uncertainty principle, while
working on the mathematical foundations of quantum mechanics. On 23 February, Heisenberg wrote a letter to
fellow physicist Wolfgang Pauli, in which he first described his new principle. In his paper on the uncertainty
principle, Heisenberg used the word "Ungenauigkeit" (imprecision).

In 1927, Heisenberg was appointed ordentlicher Professor (ordinarius professor) of theoretical physics and head of
the department of physics at the Universitat Leipzig; he gave his inaugural lecture on 1 February 1928. In his first

paper published from Leipzig, Heisenberg used the Pauli exclusion principle to solve the mystery of

t t - [3] [4] [18] [21]

ferromagnetism.

In Heisenberg's tenure at Leipzig, the quality of doctoral students, post-graduate and research associates who studied
and worked with Heisenberg there is attested to by the acclaim later earned by these people; at various times, they
included: Erich Bagge, Felix Bloch, Ugo Fano, Siegfried Fliigge, William Vermillion Houston, Friedrich Hund,
Robert S. Mulliken, Rudolf Peierls, George Placzek, Isidor Isaac Rabi, Fritz Sauter, John C. Slater, Edward Teller,
John Hasbrouck van Vleck, Victor Frederick Weisskopf, Carl Friedrich von Weizsacker, Gregor Wentzel and
Clarence Zener.

[231 [241

In early 1929, Heisenberg and Pauli submitted the first of two papers laying the foundation for relativistic

quantum field theory. Also in 1929, Heisenberg went on a lecture tour in the United States, Japan, China, and
India. [18] [22]

Shortly after the discovery of the neutron by James Chadwick in 1932, Heisenberg submitted the first of three
paps

[28]

papers on his neutron-proton model of the nucleus. He was awarded the 1932 Nobel Prize in Physics.

In 1928, the British mathematical physicist P. A. M. Dirac had derived the relativistic wave equation of quantum
mechanics, which implied the existence of positive electrons, later to be named positrons. In 1932, from a cloud
chamber photograph of cosmic rays, the American physicist Carl David Anderson identified a track as having been
made by a positron. In mid-1933, Heisenberg presented his theory of the positron. His thinking on Dirac's theory and

further development of the theory were set forth in two papers. The first, Bemerkungen zur Diracschen Theorie des

[291
Positrons (Remarks on Dirac's theory of the positron) was published in 1934, and the second, Folgerungen aus

n 8i
der Diracschen Theorie des Positrons (Consequences of Dirac's Theory of the Positron), was published in 1936.

.In these papers Heisenberg was the first to reinterpret the Dirac equation as a "classical" field equation for

any point particle of spin fi/2, itself subject to quantization conditions involving anti-commutators. Thus

reinterpreting it as a (quantum) field equation accurately describing electrons, Heisenberg put matter on the same

footing as electromagnetism: as being described by relativistic quantum field equations which allowed the possibility

of particle creation and destruction.

In the early 1930s in Germany, the deutsche Physik movement was anti-Semitic and anti -theoretical physics,
especially including quantum mechanics and the theory of relativity. As applied in the university environment,

Werner Heisenberg 503

[32]
political factors took priority over the historically applied concept of scholarly ability, even though its two most

[331 [34]

prominent supporters were the Nobel Laureates in Physics Philipp Lenard and Johannes Stark.

[35]
After Adolf Hitler came to power in 1933, Heisenberg was attacked in the press as a "White Jew" by elements of

the deutsche Physik (German Physics) movement for his insistence on teaching about the roles of Jewish scientists.

As a result, he came under investigation by the SS. This was over an attempt to appoint Heisenberg as successor to

Arnold Sommerfeld at the University of Munich. The issue was resolved in 1938 by Heinrich Himmler, head of the

SS. While Heisenberg was not chosen as Sommerfeld's successor, he was rehabilitated to the physics community

during the Third Reich. Nevertheless, supporters of deutsche Physik launched vicious attacks against leading

theoretical physicists, including Arnold Sommerfeld and Heisenberg. On 29 June 1936, a National Socialist Party

newspaper published a column attacking Heisenberg. On 15 July 1937, he was attacked in a journal of the SS. This

[IB]

was the beginning of what is called the Heisenberg Affair.

In mid-1936, Heisenberg presented his theory of cosmic -ray showers in two papers. Four more papers

[40] [41] , . ., . , [18] [42]

appeared in the next two years.

In June 1939, Heisenberg bought a summer home for his family in Urfeld, in southern Germany. He also traveled to
the United States in June and July, visiting Samuel Abraham Goudsmit, at the University of Michigan in Ann Arbor.
However, Heisenberg refused an invitation to emigrate to the United States. He did not see Goudsmit again until six
years later, when Goudsmit was the chief scientific advisor to the American Operation Alsos at the close of World
War II. Ironically, Heisenberg was arrested under Operation Alsos and detained in England under Operation
Epsilon. [18] [43] [44]

Matrix Mechanics and the Nobel Prize

i [45]

Heisenberg s paper establishing quantum mechanics has puzzled
physicists and historians. His methods assume that the reader is
familiar with Kramers-Heisenberg transition probability calculations.
The main new idea, noncommuting matrices, is justified only by a
rejection of unobservable quantities. It introduces the non-commutative
multiplication of matrices by physical reasoning, based on the
correspondence principle, despite the fact that Heisenberg was not then
familiar with the mathematical theory of matrices. The path leading to
these results has been reconstructed in MacKinnon, 1977, and the
detailed calculations are worked out in Aitchison et al. [47] Niels Bohr - Wemer Heisenberg, and Wolfgang

Pauli, ca. 1935

In Copenhagen, Heisenberg and H. Kramers collaborated on a paper on
dispersion, or the scattering from atoms of radiation whose wavelength is larger than the atoms. They showed that
the successful formula Kramers had developed earlier could not be based on Bohr orbits, because the transition
frequencies are based on level spacings which are not constant. The frequencies which occur in the Fourier transform
of sharp classical orbits, by contrast, are equally spaced. But these results could be explained by a semi-classical
Virtual State model: the incoming radiation excites the valence, or outer, electron to a virtual state from which it
decays. In a subsequent paper Heisenberg showed that this virtual oscillator model could also explain the
polarization of fluorescent radiation.

These two successes, and the continuing failure of the Bohr-Sommerfeld model to explain the outstanding problem
of the anomalous Zeeman effect, led Heisenberg to use the virtual oscillator model to try to calculate spectral
frequencies. The method proved too difficult to immediately apply to realistic problems, so Heisenberg turned to a
simpler example, the anharmonic oscillator.

The dipole oscillator consists of a simple harmonic oscillator, which is thought of as a charged particle on a spring,
perturbed by an external force, like an external charge. The motion of the oscillating charge can be expressed as a

Werner Heisenberg 504

Fourier series in the frequency of the oscillator. Heisenberg solved for the quantum behavior by two different
methods. First, he treated the system with the virtual oscillator method, calculating the transitions between the levels
that would be produced by the external source.

He then solved the same problem by treating the anharmonic potential term as a perturbation to the harmonic
oscillator and using the perturbation methods that he and Born had developed. Both methods led to the same results
for the first and the very complicated second order correction terms. This suggested that behind the very complicated
calculations lay a consistent scheme.

So Heisenberg set out to formulate these results without any explicit dependence on the virtual oscillator model. To
do this, he replaced the Fourier expansions for the spatial coordinates by matrices, matrices which corresponded to
the transition coefficients in the virtual oscillator method. He justified this replacement by an appeal to Bohr's
correspondence principle and the Pauli doctrine that quantum mechanics must be limited to observables.

On 9 July, Heisenberg gave Born this paper to review and submit for publication. When Born read the paper, he

recognized the formulation as one which could be transcribed and extended to the systematic language of

T481 T491

matrices, which he had learned from his study under Jakob Rosanes at Breslau University. Born, with the help

of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and

they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg's

paper. A follow-on paper was submitted for publication before the end of the year by all three authors. (A

brief review of Born's role in the development of the matrix mechanics formulation of quantum mechanics along

with a discussion of the key formula involving the non-commutivity of the probability amplitudes can be found in an

T521
article by Jeremy Bernstein, Max Born and the Quantum Theory. A detailed historical and technical account can

be found in Mehra and Rechenberg's book The Historical Development of Quantum Theory. Volume 3. The

T531
Formulation of Matrix Mechanics and Its Modifications 1925—1926. )

Up until this time, matrices were seldom used by physicists; they were considered to belong to the realm of pure
mathematics. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work

on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their

T541
multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics.

Born had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert's theory of
integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of
Hilbert's work Grundziige einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912.
Jordan, too was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at

Gottingen in the preparation of Courant and David Hilbert's book Methoden der mathematischen Physik I, which was

T571
published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the

continued development of quantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and

he coined the term Hilbert space to describe the algebra and analysis which were used in the development of

quantum mechanics.

In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, but it was not to
be. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933. It was at that
time that it was announced Heisenberg had won the Prize for 1932 "for the creation of quantum mechanics, the
application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen" and Erwin

Schrodinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of
atomic theory". One can rightly ask why Born was not awarded the Prize in 1932 along with Heisenberg —
Bernstein gives some speculations on this matter. One of them is related to Jordan joining the Nazi Party on 1 May
1933 and becoming a Storm Trooper. Hence, Jordan's Party affiliations and Jordan's links to Born may have
affected Born's chance at the Prize at that time. Bernstein also notes that when Born won the Prize in 1954, Jordan
was still alive, and the Prize was awarded for the statistical interpretation of quantum mechanics, attributable alone
to Born.

Werner Heisenberg 505

Heisenberg's reaction to Born for Heisenberg receiving the Prize for 1932 and to Born for Born receiving the Prize in
1954 are also instructive in evaluating whether Born should have shared the Prize with Heisenberg. On 25 November
1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad
conscience" that he alone had received the Prize "for work done in Gottingen in collaboration — you, Jordan and I."
Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a
wrong decision from the outside." In 1954, Heisenberg wrote an article honoring Max Planck for his insight in
1900. In the article, Heisenberg credited Born and Jordan for the final mathematical formulation of matrix mechanics
and Heisenberg went on to stress how great their contributions were to quantum mechanics, which were not
"adequately acknowledged in the public eye."

The deutsche Physik movement

On 1 April 1935, the eminent theoretical physicist Arnold Sommerfeld, Heisenberg's doctoral advisor at the
University of Munich, achieved emeritus status. However, Sommerfeld stayed in his chair during the selection
process for his successor, which took until 1 December 1939. The process was lengthy due to academic and political
differences between the Munich Faculty's selection and that of the Reichserziehungsministerium (REM, Reich
Education Ministry.) and the supporters of Deutsche Physik, which was anti-Semitic and had a bias against
theoretical physics, especially quantum mechanics and the theory of relativity. In 1935, the Munich Faculty drew up
a list of candidates to replace Sommerfeld as ordinarius professor of theoretical physics and head of the Institute for
Theoretical Physics at the University of Munich. There were three names on the list: Werner Heisenberg, who
received the Nobel Prize in Physics for 1932, Peter Debye, who received the Nobel Prize in Chemistry in 1936, and
Richard Becker - all former students of Sommerfeld. The Munich Faculty was firmly behind these candidates, with
Heisenberg as their first choice. However, supporters of Deutsche Physik and elements in the REM had their own list
of candidates and the battle dragged on for over four years. During this time, Heisenberg came under vicious attack
by the Deutsche Physik supporters. One attack was published in Das Schwarze Korps, the newspaper of the
Schutzstaffel (SS), headed by Heinrich Himmler. In this, Heisenberg was called a "White Jew" (i.e. an Aryan who
acts like a Jew) who should be made to "disappear". These attacks were taken seriously, as Jews were violently
attacked and incarcerated. Heisenberg fought back with an editorial and a letter to Himmler, in an attempt to resolve
this matter and regain his honour. At one point, Heisenberg's mother visited Himmler's mother. The two women
knew each other as Heisenberg's maternal grandfather and Himmler's father were rectors and members of a Bavarian
hiking club. Eventually, Himmler settled the Heisenberg affair by sending two letters, one to SS Gruppenfuhrer
Reinhard Heydrich and one to Heisenberg, both on 21 July 1938. In the letter to Heydrich, Himmler said Germany
could not afford to lose or silence Heisenberg as he would be useful for teaching a generation of scientists. To
Heisenberg, Himmler said the letter came on recommendation of his family and he cautioned Heisenberg to make a
distinction between professional physics research results and the personal and political attitudes of the involved
scientists. The letter to Heisenberg was signed under the closing "Mit freundlichem Gruss und, Heil Hitler!" (With
friendly greetings, Heil Hitler!") Overall, the Heisenberg affair was a victory for academic standards and
professionalism. However, the appointment of Wilhelm Mtiller to replace Sommerfeld was a political victory over
academic standards. Miiller was not a theoretical physicist, had not published in a physics journal, and was not a
member of the Deutsche Physikalische Gesellschaft; his appointment was considered a travesty and detrimental to

a *• t u <•• i v. • • * [69] [70] [71] [72] [73]

educating theoretical physicists.

During the SS investigation of Heisenberg, the three investigators had training in physics. Heisenberg had
participated in the doctoral examination of one of them at the Universitdt Leipzig- The most influential of the three,
however, was Johannes Juilfs. During their investigation, they had become supporters of Heisenberg as well as his
position against the ideological policies of the deutsche Physik movement in theoretical physics and academia.

Werner Heisenberg 506

World War II

In 1939, shortly after the discovery of nuclear fission, the German nuclear energy project, also known as the
Uranverein (Uranium Club), was begun. Heisenberg was one of the principal scientists leading research and
development in the project.

From 15 to 22 September 1941, Heisenberg traveled to German-occupied Copenhagen to lecture and discuss nuclear
research and theoretical physics with Niels Bohr. The meeting, and specifically what it might reveal about
Heisenberg's intentions concerning developing nuclear weapons for the Nazi regime, is the subject of the award
winning Michael Frayn play titled Copenhagen. Documents relating to the Bohr-Heisenberg meeting were released

r-7C"i [""7A"]

in 2002 by the Niels Bohr Archive and by the Heisenberg family.

On 26 February 1942, Heisenberg presented a lecture to Reich officials on energy acquisition from nuclear fission,

T771
after the Army withdrew most of its funding. The Uranium Club was transferred to the Reich Research Council

(RFR) in July 1942. On 4 June 1942, Heisenberg was summoned to report to Albert Speer, Germany's Minister of

Armaments, on the prospects for converting the Uranium Club's research toward developing nuclear weapons.

During the meeting, Heisenberg told Speer that a bomb could not be built before 1945, and would require significant

T781

monetary and manpower resources. Five days later, on 9 June 1942, Adolf Hitler issued a decree for the

reorganization of the RFR as a separate legal entity under the Reich Ministry for Armament and Ammunition; the

[791
decree appointed Reich Marshall Goring as the president.

In September 1942, Heisenberg submitted his first paper of a three-part series on the scattering matrix, or S -matrix,

rsoi rsii rs?i

in elementary particle physics. The first two papers were published in 1943 and the third in 1944. The

S-matrix described only observables, i.e., the states of incident particles in a collision process, the states of those

emerging from the collision, and stable bound states; there would be no reference to the intervening states. This was

the same precedent as he followed in 1925 in what turned out to be the foundation of the matrix formulation of

quantum mechanics through only the use of observables.

In February 1943, Heisenberg was appointed to the Chair for Theoretical Physics at the
Friedrich-Wilhelms-Universitdt (today, the Humboldt-Universitat zu Berlin). In April, his election to the Preufiische
Akademie der Wissenschaften (Prussian Academy of Sciences) was approved. That same month, he moved his
family to their retreat in Urfeld as Allied bombing increased in Berlin. In the summer, he dispatched the first of his
staff at the Kaiser-Wilhelm Institut fur Physik to Hechingen and its neighboring town of Haigerloch, on the edge of

the Black Forest, for the same reasons. From 18—26 October, he traveled to German-occupied Netherlands. In

n 8i rs^i
December 1943, Heisenberg visited German-occupied Poland.

From 24 January to 4 February 1944, Heisenberg traveled to occupied Copenhagen, after the German Army
confiscated Bohr's Institute of Theoretical Physics. He made a short return trip in April. In December, Heisenberg

no]

lectured in neutral Switzerland.

In January 1945, Heisenberg vacated the Kaiser-Wilhelm Institut fur Physik with about all of his staff for the

n si
facilities in the Black Forest.

Uranium Club

In December 1938, the German chemists Otto Hahn and Fritz Strassmann sent a manuscript to Naturwissenschaften
reporting they had detected the element barium after bombarding uranium with neutrons; simultaneously, they
communicated these results to Lise Meitner, who had in July of that year fled to the Netherlands and then went to

roc]

Sweden. Meitner, and her nephew Otto Robert Frisch, correctly interpreted these results as being nuclear

fission. Frisch confirmed this experimentally on 13 January 1939.

[88]

Paul Harteck was director of the physical chemistry department at the University of Hamburg and an advisor to the
Heereswaffenamt (HWA, Army Ordnance Office). On 24 April 1939, along with his teaching assistant Wilhelm
Groth, Harteck made contact with the Reichskriegsministerium (RKM, Reich Ministry of War) to alert them to the
potential of military applications of nuclear chain reactions. Two days earlier, on 22 April 1939, after hearing a

Werner Heisenberg 507

colloquium paper by Wilhelm Hanle on the use of uranium fission in a Uranmaschine (uranium machine, i.e.,
nuclear reactor), Georg Joos, along with Hanle, notified Wilhelm Dames, at the Reichserziehungsministerium (REM,
Reich Ministry of Education), of potential military applications of nuclear energy. The communication was given to
Abraham Esau, head of the physics section of the Reichsforschungsrat (RFR, Reich Research Council) at the REM.
On 29 April, a group, organized by Esau, met at the REM to discuss the potential of a sustained nuclear chain
reaction. The group included the physicists Walther Bothe, Robert Dopel, Hans Geiger, Wolfgang Gentner (probably
sent by Walther Bothe), Wilhelm Hanle, Gerhard Hoffmann and Georg Joos; Peter Debye was invited, but he did not
attend. After this, informal work began at the Georg-August University of Gottingen by Joos, Hanle and their
colleague Reinhold Mannfopff; the group of physicists was known informally as the first Uranverein (Uranium
Club) and formally as Arbeitsgemeinschaft fur Kernphysik. The group's work was discontinued in August 1939,
when the three were called to military training.

The second Uranverein began after the Heereswaffenamt (HWA, Army Ordnance Office) squeezed the
Reichsforschungsrat (RFR, Reich Research Council) out of the Reichserziehungsministerium (REM, Reich Ministry
of Education) and started the formal German nuclear energy project under military auspices. The second Uranverein
was formed on 1 September 1939, the day World War II began, and it had its first meeting on 16 September 1939.
The meeting was organized by Kurt Diebner, advisor to the HWA, and held in Berlin. The invitees included Walther
Bothe, Siegfried Flugge, Hans Geiger, Otto Hahn, Paul Harteck, Gerhard Hoffmann, Josef Mattauch and Georg
Stetter. A second meeting was held soon thereafter and included Klaus Clusius, Robert Dopel, Werner Heisenberg
and Carl Friedrich von Weizsacker. Also at this time, the Kaiser-Wilhelm Institutfur Physik (KWIP, Kaiser Wilhelm
Institute for Physics, after World War II the Max Planck Institute for Physics), in Berlin-Dahlem, was placed under
HWA authority, with Diebner as the administrative director, and the military control of the nuclear research
commenced.

When it was apparent that the nuclear energy project would not make a decisive contribution to ending the war effort
in the near term, control of the KWIP was returned in January 1942 to its umbrella organization, the Kaiser-Wilhelm
Gesellschaft (KWG, Kaiser Wilhelm Society, after World War II the Max-Planck Gesellschaft), and HWA control of
the project was relinquished to the RFR in July 1942. The nuclear energy project thereafter maintained its
kriegswichtig (important for the war) designation and funding continued from the military. However, the German
nuclear power project was then broken down into the following main areas: uranium and heavy water production,
uranium isotope separation and the Uranmaschine (uranium machine, i.e., nuclear reactor). Also, the project was
then essentially split up between a number of institutes, where the directors dominated the research and set their own
research agendas. The dominant personnel and facilities were the following:

• Institutfur Physik (Walther Bothe) of the Kaiser-Wilhelm Institutfur medizinische Forschung (KWImF, Kaiser
Wilhelm Institute for Medical Research),

Institute for Physical Chemistry (Klaus Clusius) at the Ludwig Maximilian University of Munich,
HWA Versuchsstelle (testing station) in Gottow (Kurt Diebner),
Kaiser -Wilhelm-Institut fur Chemie (Otto Hahn),

Physical Chemistry Department (Paul Harteck) of the University of Hamburg,
Kaiser-Wilhelm-Institut fur Physik (Werner Heisenberg),

Second Experimental Physics Institute (Hans Kopfermann) at the Georg-August University of Gottingen,
Auergesellschaft (Nikolaus Riehl), and
77. Physikalisches Institut (Georg Stetter) at the University of Vienna.

Heisenberg was appointed director-in-residence of the KWIP on 1 July 1942, as Peter Debye was still officially the
director and on leave in the United States; Debye had gone on leave as he was a citizen of The Netherlands and had
refused to become a German citizen when the HWA took administrative control of the KWIP. Heisenberg still also
had his department of physics at the University of Leipzig where work was done for the Uranverein by Robert Dopel
and his wife Klara Dopel. During the period Kurt Diebner administered the KWIP under the HWA program,

Werner Heisenberg 508

considerable personal and professional animosity developed between Diebner and the Heisenberg inner circle —
Heisenberg, Karl Wirtz, and Carl Friedrich von Weizsacker.

The point in 1942, when the army relinquished its control of the German nuclear energy project, was the zenith of
the project relative to the number of personnel devoting time to the effort. There were only about 70 scientists
working on the project, with about 40 devoting more than half their time to nuclear fission research. After this, the
number of scientists working on applied nuclear fission diminished dramatically. Many of the scientists not working
with the main institutes stopped working on nuclear fission and devoted their efforts to more pressing war related
work.

Over time, the HWA and then the RFR controlled the German nuclear energy project. The most influential people in
the project were Kurt Diebner, Abraham Esau, Walther Gerlach and Erich Schumann. Schumann was one of the
most powerful and influential physicists in Germany. Schumann was director of the Physics Department II at the
Frederick William University (later, University of Berlin), which was commissioned and funded by the
Oberkommando des Heeres (OKW, Army High Command) to conduct physics research projects. He was also head
of the research department of the HWA, assistant secretary of the Science Department of the OKW and
Bevollmdchtiger (plenipotentiary) for high explosives. Diebner, throughout the life of the nuclear energy project, had
more control over nuclear fission research than did Walther Bothe, Klaus Clusius, Otto Hahn, Paul Harteck or
Werner Heisenberg.

1945: Operation Alsos and Operation Epsilon

Operation Alsos was an Allied effort commanded by the Russian-American Colonel Boris T. Pash. He reported
directly to General Leslie Groves, commander of the Manhattan Engineer District, which was developing atomic
weapons for the United States. The chief scientific advisor to Operation Alsos was the physicist Samuel Abraham
Goudsmit. Goudsmit was selected for this task because of his knowledge of physics, he spoke German, and he
personally knew a number of the German scientists working on the German nuclear energy project. He also knew
little of the Manhattan Project, so, if he were captured, he would have little intelligence value to the Germans. The
objectives of Operation Alsos were to determine if the Germans had an atomic bomb program and to exploit German
atomic related facilities, intellectual materials, materiel resources, and scientific personnel for the benefit of the
United States. Personnel on this operation generally swept into areas which had just come under control of the Allied
military forces, but sometimes they operated in areas still under control by German forces.

Berlin had been a location of many German scientific research facilities. To limit casualties and loss of equipment,
many of these facilities were dispersed to other locations in the latter years of the war. The Kaiser-Wilhelm-Institut
fiir Physik (KWIP, Kaiser Wilhelm Institute for Physics) had mostly been moved in 1943 and 1944 to Hechingen and
its neighboring town of Haigerloch, on the edge of the Black Forest, which eventually became the French occupation
zone. This move and a little luck allowed the Americans to take into custody a large number of German scientists
associated with nuclear research. The only section of the institute which remained in Berlin was the low-temperature
physics section, headed by Ludwig Bewilogua (1906—83), who was in charge of the exponential uranium pile.

[107]

Nine of the prominent German scientists who published reports in Kernphysikalische Forschungsberichte as
members of the Uranverein were picked up by Operation Alsos and incarcerated in England under Operation
Epsilon: Erich Bagge, Kurt Diebner, Walther Gerlach, Otto Hahn, Paul Harteck, Werner Heisenberg, Horst
Korsching, Carl Friedrich von Weizsacker and Karl Wirtz. Also, incarcerated was Max von Laue, although he had
nothing to do with the nuclear energy project. Goudsmit, the chief scientific advisor to Operation Alsos, thought von
Laue might be beneficial to the postwar rebuilding of Germany and would benefit from the high level contacts he
would have in England.

Heisenberg had been captured and arrested by Colonel Pash at Heisenberg's retreat in Urfeld, on 3 May 1945, in
what was a true alpine-type operation in territory still under control by German forces. He was taken to Heidelberg,

Werner Heisenberg 509

where, on 5 May, he met Goudsmit for the first time since the Ann Arbor visit in 1939. Germany surrendered just
two days later. Heisenberg did not see his family again for eight months. Heisenberg was moved across France and
Belgium and flown to England on 3 July 1945.

The 10 German scientists were held at Farm Hall in England. The facility had been a safe house of the British
foreign intelligence MI6. During their detention, their conversations were recorded. Conversations thought to be of
intelligence value were transcribed and translated into English. The transcripts were released in 1992. Bernstein has
published an annotated version of the transcripts in his book Hitler's Uranium Club: The Secret Recordings at Farm
Hall, along with an introduction to put them in perspective. A complete, unedited publication of the British version
of the reports appeared as Operation Epsilon: The Farm Hall Transcripts, which was published in 1993 by the
Institute of Physics in Bristol and by the University of California Press in the United States.

Post 1945

On 3 January 1946, the 10 Operation Epsilon detainees were transported to Alswede, Germany, which was in the
British occupation zone. Heisenberg settled in Gottingen, also in the British zone. In July, he was named director of
the Kaiser-Wilhelm Institut fur Physik (KWIP, Kaiser Wilhelm Institute for Physics), then located in Gottingen.
Shortly thereafter, it was renamed the Max-Planck Institut fur Physik, in honor of Max Planck and to assuage
political objections to the continuation of the institute. Heisenberg was its director until 1958. In 1958, the institute
was moved to Munich, expanded, and renamed Max-Planck-Institut fur Physik und Astrophysik (MPIFA).
Heisenberg was its director from 1960 to 1970; in the interim, Heisenberg and the astrophysicist Ludwig Biermann
were co-directors. Heisenberg resigned his directorship of the MPIFA on 31 December 1970. Upon the move to
Munich, Heisenberg also became an ordentlicher Professor (ordinarius professor) at the University of Munich.

Just as the Americans did with Operation Alsos, the Russians inserted special search teams into Germany and
Austria in the wake of their troops. Their objective, under the Russian Alsos, was also the exploitation of German
atomic related facilities, intellectual materials, materiel resources and scientific personnel for the benefit of the
Soviet Union. One of the German scientists recruited under this Russian operation was the nuclear physicist Heinz
Pose, who was made head of Laboratory V in Obninsk. When he returned to Germany on a recruiting trip for his
laboratory, Pose wrote a letter to the Werner Heisenberg inviting him to work in Russia. The letter lauded the
working conditions in Russia and the available resources, as well as the favorable attitude of the Russians towards
German scientists. A courier hand delivered the recruitment letter, dated 18 July 1946, to Heisenberg; Heisenberg
politely declined in a return letter to Pose.

In 1947, Heisenberg presented lectures in Cambridge, Edinburgh and Bristol. Heisenberg also contributed to the
understanding of the phenomenon of superconductivity with a paper in 1947 and two papers in 1948,
one of them with Max von Laue.

In the period shortly after World War II, Heisenberg briefly returned to the subject of his doctoral thesis, turbulence.
Three papers were published in 1948 and one in 1950.

In the post-war period, Heisenberg continued his interests in cosmic-ray showers with considerations on multiple
production of mesons. He published three papers in 1949, two in 1952, and one in

1955. [132]

On 9 March 1949, the Deutsche Forschungsrat (German Research Council) was established by the Max-Planck
Gesellschaft (MPG, Max Planck Society, successor organization to the Kaiser-Wilhelm Gesellschaft. Heisenberg was
appointed president of the Deutsche Forschungsrat. In 1951, the organization was fused with the Notgemeinschaft
der Deutschen Wissenschaft (NG, Emergency Association of German Science) and that same year renamed the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). With the merger, Heisenberg was
appointed to the presidium.

In 1952, Heisenberg served as the chairman of the Commission for Atomic Physics of the DFG. Also that year, he

rQ] MS]

headed the German delegation to the European Council for Nuclear Research.

Werner Heisenberg 510

In 1953, Heisenberg was appointed president of the Alexander von Humboldt-Stiftung by Konrad Adenauer.
Heisenberg served until 1975. Also, from 1953, Heisenberg's theoretical work concentrated on the unified field
theory of elementary particles.

In late 1955 to early 1956, Heisenberg gave the Gifford Lectures at St Andrews University, in Scotland, on the
intellectual history of physics. The lectures were later published as Physics and Philosophy: The Revolution in
Modern Science.

During 1956 and 1957, Heisenberg was the chairman of the Arbeitskreis Kernphysik (Nuclear Physics Working
Group) of the Fachkommission II "Forschung und Nachwuchs" (Commission II "Research and Growth") of the
Deutschen Atomkommission (DAtK, German Atomic Energy Commission). Other members of the Nuclear Physics
Working Group in both 1956 and 1957 were: Walther Bothe, Hans Kopfermann (vice-chairman), Fritz Bopp,
Wolfgang Gentner, Otto Haxel, Willibald Jentschke, Heinz Maier-Liebnitz, Josef Mattauch, Wolfgang Riezler,
Wilhelm Walcher and Carl Friedrich von Weizsacker. Wolfgang Paul was also a member of the group during

195? [136]

ri37i
In 1957, Heisenberg was a signatory of the manifesto of the Gottinger Achtzehn (Gottingen Eighteen).

From 1957, Heisenberg was interested in plasma physics and the process of nuclear fusion. He also collaborated with

the International Institute of Atomic Physics in Geneva. He was a member of the Institute's Scientific Policy

T31
Committee, and for several years was the Committee's chairman.

In 1973, Heisenberg gave a lecture at Harvard University on the historical development of the concepts of quantum
theory.

On 24 March 1973, Heisenberg gave a speech before the Catholic Academy of Bavaria, accepting the Romano
Guardini Prize. An English translation of its title is "Scientific and Religious Truth." And its stated goal was "In what
follows, then, we shall first of all deal with the unassailability and value of scientific truth, and then with the much
wider field of religion, of which - so far as the Christian religion is concerned - Guardini himself has so persuasively
written; finally - and this will be the hardest part to formulate - we shall speak of the relationship of the two
truths." A more detail insight in Planck and Heisenberg on religion has been discussed by Wilfried Schroder in "
Natural science and religion" (Bremen 1999, Science edition) and Wilfried Schroder " Naturerkenntnis und
Religion" Bremen, science edition 2008).

Personal life

In January 1937 Heisenberg met Elisabeth Schumacher at a private music recital. Elisabeth was the daughter of a
well-known Berlin economics professor. They were married on 29 April. The fraternal twins, Maria and Wolfgang,
were born to them in January 1938, whereupon, Wolfgang Pauli congratulated Heisenberg on his "pair creation" — a
word play on a process from elementary particle physics, pair production. They had five more children over the next
12 years: Barbara, Christine, Jochen, Martin and Verena. Jochen became a physics professor at the University of

m tr u- [140] [141]

New Hampshire.

T31 ri421

Heisenberg enjoyed classical music and was an accomplished pianist. He was a Lutheran Christian.

Heisenberg died of cancer of the kidneys and gall bladder at his home, on 1 February 1976. The next evening,
his colleagues and friends walked in remembrance from the Institute of Physics to his home and each put a candle
near the front door.

Werner Heisenberg 511

Honors and awards

Heisenberg was awarded a number of honors:

• Honorary doctorates from the University of Bruxelles, the Technological University of Karlsruhe, and the
University of Budapest.

• Order of Merit of Bavaria

• Romano Guardini Prize

• Grand Cross for Federal Service with Star

• Knight of the Order of Merit (Peace Class)

• Fellow of the Royal Society of London

• Member of the Academies of Sciences of Gottingen, Bavaria, Saxony, Prussia, Sweden, Rumania, Norway,
Spain, The Netherlands, Rome (Pontifical), the Deutsche Akademie der Naturforscher Leopoldina (Halle), the
Accademia dei Lincei (Rome), and the American Academy of Sciences.

• 1932— Nobel Prize in Physics "for the creation of quantum mechanics, the application of which has, inter alia, led
to the discovery of the allotropic forms of hydrogen".

• 1933- Max-Planck-Medaille of the Deutsche Physikalische Gesellschaft

Internal reports

The following reports were published in Kernphysikalische Forschungsberichte {Research Reports in Nuclear
Physics), an internal publication of the German Uranverein. The reports were classified Top Secret, they had very
limited distribution, and the authors were not allowed to keep copies. The reports were confiscated under the Allied
Operation Alsos and sent to the United States Atomic Energy Commission for evaluation. In 1971, the reports were
declassified and returned to Germany. The reports are available at the Karlsruhe Nuclear Research Center and the
American Institute of Physics.

Robert Dopel, K. Dopel, and Werner Heisenberg Bestimmung der Diffusionsldnge thermischer Neutronen in

Praparat 38 [U7] G-22 (5 December 1940)

Robert Dopel, K. Dopel, and Werner Heisenberg Bestimmung der Diffusionsldnge thermischer Neutronen in

schwerem Wasser G-23 (7 August 1940)

Werner Heisenberg Die Moglichkeit der technischer Energiegewinnung aus der Uranspaltung G-39 (6 December

1939)

Werner Heisenberg Bericht ilber die Moglichkeit technischer Energiegewinnung aus der Uranspaltung (II) G-40

(29 February 1940)

Robert Dopel, K. Dopel, and Werner Heisenberg Versuche mit Schichtenanordnungen von DO und 38 G-75 (28

October 1941)

Werner Heisenberg Uber die Moglichkeit der Energieerzeugung mit Hilfe des Isotops 238 G-92 (1941)

Werner Heisenberg Bericht ilber Versuche mit Schichtenanordnungen von Praparat 38 und Paraffin am Kaiser

Wilhelm Institutfilr Physik in Berlin-Dahlem G-93 (May 1941)

Fritz Bopp, Erich Fischer, Werner Heisenberg, Carl-Friedrich von Weizsacker, and Karl Wirtz Untersuchungen

mit neuen Schichtenanordnungen aus U-metall und Paraffin G-127 (March 1942)

Robert Dopel Bericht ilber Unfalle beim Umgang mit Uranmetall G-135 (9 July 1942)

Werner Heisenberg Bemerkungen zu dem geplanten halbtechnischen Versuch mit 1,5 to DO und 3 to 38-Metall

G- 161 (31 July 1942)

Werner Heisenberg, Fritz Bopp, Erich Fischer, Carl-Friedrich von Weizsacker, and Karl Wirtz Messungen an

Schichtenanordnungen aus 38-Metall und Paraffin G-162 (30 October 1942)

Robert Dopel, K. Dopel, and Werner Heisenberg Der experimentelle Nachweis der effektiven

Neutronenvermehrung in einem Kugel-Schichten-System aus DO und Uran-Metall G-136 (July 1942)

Werner Heisenberg Die Energiegewinnung aus der Atomkernspaltung G-217 (6 May 1943)

Werner Heisenberg 512

• Fritz Bopp, Walther Bothe, Erich Fischer, Erwin Filnfer, Werner Heisenberg, O. Ritter, and Karl Wirtz Bericht
tiber einen Versuch mit 1.5 to DO und U und 40 cm Kohlerilckstreumantel (B7) G-300 (3 January 1945)

• Robert Dopel, K. Dopel, and Werner Heisenberg Die Neutronenvermehrung in einem
D 2 0-38-Metallschichtensystem G-373 (March 1942)

Publications

Collected bibliographies

Cassidy, David C. Werner Heisenberg : A Bibliography of His Writings, Second, Expanded Edition (Whittier,
2001)

Cassidy, David Werner Heisenberg: A Bibliography of His Writings, 1922—1929, Expanded Edition HTML
Version PDF Version

Mott, N. and R. Peierls Werner Heisenberg, Biographical Memoirs of Fellows of the Royal Society Volume 23,
213-251 (1977)

Anna Ludovico, Effetto Heisenberg. La rivoluzione scientifica che ha cambiato la storia, Roma: Armando 2001,
p. 224 ISBN 88-8358-182-2.

Barbara Blum, Helmut Heisenberg, Anna Ludovico, Per Heisenberg, Roma: Aracne 2006, p. 96 ISBN
88-548-0636-6
Selected articles

A. Sommerfeld and W. Heisenberg Eine Bemerkung tiber relativistische Rontgendubletts und Linienschdrfe, Z.
Phys. Volume 10, 393-398 (1922)

A. Sommerfeld and W. Heisenberg Die Intensitat der Mehrfachlinien und ihrer Zeeman-Komponenten, Z. Phys.
Volume 11, 131-154(1922)

M. Born and W. Heisenberg Uber Phasenbeziehungen bei den Bohrschen Modellen von Atomen und Molekeln, Z.
Phys. Volume 14, 44-55 (1923)

M. Born and W. Heisenberg Die Elektronenbahnen im angeregten Heliumatom, Z. Phys. Volume 16, 229-243
(1923)

M. Born and W. Heisenberg Zur Quantentheorie der Molekeln, Ann. d. Physik Volume 74, Number 4, 1-31
(1924)

W. Heisenberg Uber Stabilitdt und Turbulent von Fltissigkeitsstrommen (Diss.), Ann. Physik Volume 74, Number
4,577-627(1924)

M. Born and W. Heisenberg Uber den Einfluss der Deformierbarekit der Ionen auf optische und chemische
Konstanten. I., Z. Phys. Volume 23, 388-410 (1924)

W. Heisenberg Uber eine Abdnderung der formalin Regeln der Quantentheorie beim Problem der anomalen
Zeeman-Effekte, Z. Phys. Volume 26, 291-307 (1924)

W. Heisenberg, Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift
ftir Physik, 33, 879-893 (1925). The paper was received on 29 July 1925. [English translation in: B. L. van der
Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title:
Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations).] This is the first paper in the
famous trilogy which launched the matrix mechanics formulation of quantum mechanics.

M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift ftir Physik, 34, 858-888 (1925). The paper was received
on 27 September 1925. [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics
(Dover Publications, 1968) ISBN 0-486-61881-1 (English title: On Quantum Mechanics).] This is the second
paper in the famous trilogy which launched the matrix mechanics formulation of quantum mechanics.

Werner Heisenberg 513

• M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift filr Physik, 35, 557-615 (1925). The
paper was received on 16 November 1925. [English translation in: B. L. van der Waerden, editor, Sources of
Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1] This is the third paper in the famous
trilogy which launched the matrix mechanics formulation of quantum mechanics.

W. Heisenberg Uber den anschulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys.
Volume 43, 172-198(1927)

W. Heisenberg Zur Theorie des Ferromagnetismus, Z. Phys. Volume 49, 619-636 (1928)

W. Heisenberg and W. Pauli Zur Quantentheorie der Wellenf elder, Z. Phys. Volume 56, 1-61 (1929)

W. Heisenberg and W. Pauli Zur Quantentheorie der Wellenf elder. II, Z. Phys. Volume 59, 168-190 (1930)

W. Heisenberg Uber den Bau der Atomkerne. I, Z. Phys. Volume 77, 1-11 (1932)

W. Heisenberg Uber den Bau der Atomkerne. II, Z. Phys. Volume 78, 156-164 (1932)

W. Heisenberg Uber den Bau der Atomkerne. Ill, Z. Phys. Volume 80, 587-596 (1933)

W. Heisenberg Bemerkungen zur Diracschen Theorie des Positrons, Zeitschrift filr Physik Volume 90, Numbers
3-4, 209-231 (1934). The author was cited as being at Leipzig. The paper was received on 21 June 1934.

W. Heisenberg Uber die 'Schauer' in der Kosmischen Strahlung, Forsch. Fortscher. Volume 12, 341-342 (1936)

W. Heisenberg and H. Euler Folgerungen aus der Diracschen Theorie des Positrons, Zeitschr. Phys. Volume 98,
Numbers 11-12, 714-732 (1936). The authors were cited as being at Leipzig. The paper was received on 22
December 1935. A translation of this paper has been done by W. Korolevski and H. Kleinert:
arXiv:physics/0605038vl [150] .

W. Heisenberg Zur Theorie der 'Schauer' in der Hohenstrahlung, Z. Phys. Volume 101, 533-540 (1936)

W. Heisenberg Der Durchgang sehr energiereicher Korpuskeln durch den Atomkern, Ber. Sachs, Akad. Wiss.
Volume 89, 369; Die Naturwissenschaften Volume 25, 749-750 (1937)

W. Heisenberg Theoretische Untersuchungen zur Ultrastrahlung, Verh. Stsch. physical. Ges. Volume 18, 50
(1937)

W. Heisenberg Die Absorption der durchdringenden Komponente der Hohenstrahlung, Ann. Phys. Volume 33,
594-599 (1938)

W. Heisenberg Der Durchgang sehr energiereicher Korpuskeln durch den Atomkern, Nuovo Cimento Volume 15,
31-34; Verh. Dtsch. physik. Ges. Volume 19, 2 (1938)

W. Heisenberg Die beobachtbaren Grossen in der Theorie der Elementarteilchen. I, Z. Phys. Volumne 120,
513-538(1943)

W. Heisenberg Die beobachtbaren Grossen in der Theorie der Elementarteilchen. II, Z. Phys. Volumne 120,
673-702 (1943)

W. Heisenberg Die beobachtbaren Grossen in der Theorie der Elementarteilchen. III., Z. Phys. Volumne 123,
93-112(1944)

W. Heisenberg Zur Theorie der Supraleitung, Forsch. Fortschr. Volumes 21/23, 243-244 (1947); Z. Naturf
Volume 2a, 185-201 (1947)

W. Heisenberg Das elektrodynamische Verhalten der Supraleiter, Z. Naturf. Volume 3a, 65-75 (1948)

M. von Laue and W. Heisenberg Das Barlowsche Rad aus supraleitendem Material, Z. Phys. Volume 124,

514-518(1948)

W. Heisenberg Zur statistischen Theorie der Turbulenz, Z. Phys. Volume 124, 628-657 (1948)

W. Heisenberg On the theory of statistical and isotropic turbulence, Proc. R. Soc. London A Volume 195,
402-406 (1948)

Werner Heisenberg 514

W. Heisenberg Bemerkungen um Turbulenzproblem, Z. Naturf. Volume 3a, 434-437 (1948)

W. Heisenberg Production of mesons showers, Nature, Lond. Volume 164, 65-67 (1949)

W. Heisenberg Die Erzeugung von Mesonen in Vielfachprozessen, Nuovo Cimento Volume 6 (Supplement),
493-497 (1949)

W. Heisenberg Uber die Entstehung von Mesonen in Vielfachprozessen, Z. Phys. Volume 126, 569-582 (1949)

W. Heisenberg On the stability of laminar flow , Proc. International Congress Mathematicians Volume II,
292-296 (1950)

W. Heisenberg Bermerkungen zur Theorie der Vielfacherzeugung von Mesonen, Die Naturwissenschaften
Volume 39, 69 (1952)

W. Heisenberg Mesonenerzeugung als Stosswellenproblem, Z. Phys. Volume 133, 65-79 (1952)

W. Heisenberg The production of mesons in very high energy collisions, Nuovo Cimento Volume 12, Supplement,
96-103 (1955)

W. Heisenberg Development of concepts in the history of quantum theory, American Journal of Physics Volume
43, Number 5, 389-394 (1975). The substance of this article was presented by Heisenberg in a lecture at Harvard
University.

Books

• Werner Heisenberg, Carl Eckart (translator), and F.C. Hoyt (translator) The Physical Principles of the Quantum
Theory (Dover, 1930)

Werner Heisenberg Das Naturbild der heutigen Physik (1955)
Werner Heisenberg Philosophic problems of nuclear science (Fawcett, 1966)
Werner Heisenberg Physics and Beyond: Encounters and Conversations (Harper & Row, 1971)
Werner Heisenberg and Jurgen Busche Quantentheorie und Philosophic: Vorlesungen und Aufsdtze (Reclam,
1979)

Werner Heisenberg Philosophical Problems of Quantum Physics (Ox Bow, 1979)
Werner Heisenberg Physik und Philosophic- Weltperspektiven. (Ullstein Taschenbuchvlg., 1988)
Werner Heisenberg Encounters with Einstein (Princeton University, 1989)

Werner Heisenberg and F. S. C. Northrop Physics and Philosophy: The Revolution in Modern Science (Great
Minds Series) (Prometheus, 1999)

Werner Heisenberg Der Teil und das Ganze: Gesprdche im Umkreis der Atomphysik (Piper, 2001)
Werner Heisenberg Deutsche und Judische Physik (Piper, 2002)
Werner Heisenberg Physik und Philosophic (Hirzel, 2007)

Werner Heisenberg Physics and Philosophy: The Revolution in Modern Science (Harper Perennial Modern
Classics, 2007) ( full text of 1958 version [151] )

Werner Heisenberg 515

Notes

[I] "The Nobel Prize in Physics 1932 — Werner Heisenberg" (http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-bio.
html). Nobel Prize. 1932. . Retrieved 8 June 2010.

[2] Cassidy, Uncertainty, 1992, 3.

[3] Werner Heisenberg Biography (http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-bio.html), Nobel Prize in Physics

1 932 Nobelprize.org.
[4] Hentschel and Hentschel, 1996, Appendix F; see the entry for Heisenberg.
[5] Cassidy, Uncertainty, 1992, 127 and Appendix A.
[6] Powers, 1993, 23.
[7] van der Waerden, 1968, 21.
[8] W. Heisenberg Uber Stabilitdt und Turbulenz von Flussigkeitsstrommen (Diss.), Ann. Physik Volume 74, Number 4, 577-627 (1924), as cited

in Mott and Peierls, 1977, 245.
[9] Mott and Peierls, 1977, 217.
[10] W. Heisenberg Uber eine Abanderung der formalin Regeln der Quantentheorie beim Problem der anomalen Zeeman-Effekte, Z. Phys.

Volume 26, 291-307 (1924), as cited in Mott and Peierls, 1977, 243.

[II] Mott and Peierls, 1977, 219.

[12] Maringer, Daniel. "Beruhmte Physiker: Werner Heisenberg eine Biographie-Pfadfinderzeit" (http://www.physik.tu-berlin.de/~dschm/

lect/heislek/html/pfadfinder.html) (in German). . Retrieved 2009-02-05.
[13] "Heisenberg Werner" (http://www.psfd.de/de/datenbank_mitmacher/einleitung.php, 67, Heisenberg- Werner) (in German). . Retrieved

2009-02-05.
[14] "Ein Leben filr die Jugendbewegung und Jugendseelsorger-100 Jahre Gottfried Simmerding" (http://www.kmf-net.de/files/muenchen/

Maerz2005.pdf) (in German). Rundbrief der Regionen Donau und Munchen (Gemeinschaft Katholischer Manner und Frauen im Bund

Neudeutschland-ND) 2/2005: 12. March 2005. .
[15] Helmut Raum (2008). "Die Pfadfinderbewegung im Freistaat Bayern Teil 53" (http://www.bdp-foerder-nord.de/Der Bundschuh 2.

Quartal.pdf) (in German). Der Bundschuh (Pfadfinderforderkreis Nordbayern e.V.) 2/2008: 23—24. .
[16] "February 1927: Heisenberg's Uncertainty Principle" (http://www.aps.org/publications/apsnews/200802/physicshistory.cfm). APS

News (American Physics Society) 17 (2). February 2008. .
[17] W. Heisenberg Uber den anschulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. Volume 43, 172-198 (1927),

cited in Mott and Peierls, 1977, 243.
[18] Cassidy, Uncertainty, 1992, Appendix A.
[19] Mott and Peierls, 1977, 224.

[20] W. Heisenberg Zur Theorie des Ferromagnetismus, Z. Phys. Volume 49, 619-636 (1928), as cited in Mott and Peierls, 1977, 243.
[21] Mott and Peierls, 1977,226-227.
[22] Mott and Peierls, 1977, 227.
[23] W. Heisenberg and W. Pauli Zur Quantentheorie der Wellenfelder, Z. Phys. Volume 56, 1-61 (1929), as cited in Mott and Peierls, 1977,

243.
[24] W. Heisenberg and W. Pauli Zur Quantentheorie der Wellenfelder. II, Z. Phys. Volume 59, 168-190 (1930), as cited in Mott and Peierls,

1977, 243.
[25] W. Heisenberg Uber den Ban der Atomkerne. I., Z. Phys. Volume 77, 1-11 (1932), as cited by Mott and Peierls, 1977, 244.
[26] W. Heisenberg Uber den Bau der Atomkerne. II, Z. Phys. Volume 78, 156-164 (1932), as cited by Mott and Peierls, 1977,244.
[27] W. Heisenberg Uber den Bau der Atomkerne. III., Z. Phys. Volume 80, 587-596 (1933), as cited by Mott and Peierls, 1977, 244.
[28] Mott and Peierls, 1977, 228.

[29] Werner Heisenberg Bemerkungen zur Diracschen Theorie des Positrons, Zeitschrift fur Physik Volume 90, Numbers 3-4, 209-231 (1934).
[30] W. Heisenberg and H. Euler Folgerungen aus der Diracschen Theorie des Positrons, Zeitschr. Phys. Volume 98, Numbers 11-12, 714-732

(1936). A translation of this paper has been done by W. Korolevski and H. Kleinert: arXiv:physics/0605038vl (http://arxiv.org/abs/

physics/0605038vl).
[31] Emilio Segre From X-rays to Quarks: Modern Physicists and Their Discoveries (Freeman, 1980, paperback editions).
[32] Beyerchen, 1997, 141-167.
[33] Beyerchen, 1977, 79-102.
[34] Beyerchen, 1977, 103-140.
[35] http://www.aip.org/history/heisenberg/plO.htm.
[36] W. Heisenberg Uber die 'Schauer' in der Kosmischen Strahlung, Forsch. Fortscher. Volume 12, 341-342 (1936), as cited by Mott and

Peierls, 1977, 244.
[37] W. Heisenberg Zur Theorie der 'Schauer' in der Hohenstrahlung, Z. Phys. Volume 101, 533-540 (1936), as cited by Mott and Peierls, 1977,

244.
[38] W. Heisenberg Der Durchgang sehr energiereicher Korpuskeln durch den Atomkern, Ber. Sachs, Akad. Wiss. Volume 89, 369; Die

Naturwissenschaften Volume 25, 749-750 (1937), as cited by Mott and Peierls, 1977, 244.

Werner Heisenberg

516

[39] W. Heisenberg Theoretische Untersuchungen zur Ultrastrahlung, Verh. Stsch. physical. Ges. Volume 18, 50 (1937), as cited by Mott and

Peierls, 1977, 244.
[40] W. Heisenberg Die Absorption der durchdringenden Komponente der Hbhenstrahlung, Ann. Phys. Volume 33, 594-599 (1938), as cited by

Mott and Peierls, 1977, 244.
[41] W. Heisenberg Der Durchgang sehr energiereicher Korpuskeln durch den Atomkern, Nuovo Cimento Volume 15, 31-34; Verh. Dtsch.

physik. Ges. Volume 19, 2 (1938), as cited by Mott and Peierls, 1977, 244.
[42] Mott and Peierls, 1977, 231.
[43] Hentschel and Hentschel, 1996, 387 and 387n20.
[44] Goudsmit, Alsos, 1986, picture facing p. 124.
[45] W. Heisenberg, Uber quantentheoretishe Umdeutung kinematisher und mechanischer Beziehungen, Zeitschrift fur Physik, 33, 879-893,

1925 (received 29 July 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications,

1968) ISBN 0-486-61881-1 (English title: "Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations").]
[46] MacKinnon, Edward, "Heisenberg, Models, and the Rise of Quantum Mechanics", Historical Studies in the Physical Sciences, Volume 8,

137-188(1977)
[47] Aitchison, Ian J. R., David A. MacManus and Thomas M. Snyder (November 2004), Understanding Heisenberg's 'magical' paper of July

1925: A new look at the calculational details, American Journal of Physics 72(11), 1370-1379 doi: 10. 11 19/1. 1775243

arXiv:quant-ph/0404009vl (http://arxiv.org/abs/quant-ph/0404009)
[48] Abraham Pais, Niels Bohr's Times in Physics, Philosophy, and Polity (Clarendon Press, 1991) ISBN 0-19-852049-2, pp 275 - 279.
[49] Max Born (http://nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.pdf) The Statistical Interpretation of Quantum

Mechanics, Nobel Lecture (1954)
[50] M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift fiir Physik, 34, 858-888, 1925 (received 27 September 1925). [English translation

in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]
[51] M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II , Zeitschrift fiir Physik, 35, 557-615, 1925 (received 16 November 1925).

[English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]
[52] Jeremy Bernstein Max Born and the Quantum Theory, Am. J. Phys. 73 (11) 999-1008 (2005)
[53] Mehra, Volume 3 (Springer, 2001)
[54] Jammer, 1966, pp. 206-207.
[55] van der Waerden, 1968, p. 51.
[56] The citation by Born was in Born and Jordan's paper, the second paper in the trilogy which launched the matrix mechanics formulation. See

van der Waerden, 1968, p. 351.
[57] Constance Reid Courant (Springer, 1996) p. 93.

[58] John von Neumann Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Mathematische Annalen 102 49—131 (1929)
[59] When von Neumann left Gottingen in 1932, his book on the mathematical foundations of quantum mechanics, based on Hilbert's

mathematics, was published under the title Mathematische Grundlagen der Quantenmechanik. See: Norman Macrae, John von Neumann: The

Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (Reprinted by the American

Mathematical Society, 1999) and Constance Reid, Hilbert (Springer- Verlag, 1996) ISBN 0-387-94674-8.
[60] Bernstein, 2004, p. 1004.
[61] Greenspan, 2005, p. 190.

[62] http://nobelprize.org/nobel_prizes/physics/laureates/1932/
[63] Nobel Prize in Physics and 1933 (http://nobelprize.org/nobel_prizes/physics/laureates/1933/press.html) — Nobel Prize Presentation

Speech.
[64] Bernstein, 2005, p. 1004.
[65] Bernstein, 2005, p. 1006.
[66] Greenspan, 2005, p. 191.
[67] Greenspan, 2005, pp. 285-286.
[68] Klaus Hentschel (Editor) and Ann M. Hentschel (Editorial Assistant and Translator) Physics and National Socialism: An Anthology of

Primary Sources (Birkhauser, 1996). In this book, see: Document #55 White Jews' in Science [15 July 1937] pp. 152-157.
[69] Goudsmit, Samuel A. ALSOS (Tomash Publishers, 1986) pp 117-119.
[70] Beyerchen, 1977, 153-167.
[71] Cassidy, 1992, 383-387.
[72] Powers, 1993, 40^3.
[73] Klaus Hentschel (Editor) and Ann M. Hentschel (Editorial Assistant and Translator) Physics and National Socialism: An Anthology of

Primary Sources (Birkhauser, 1996). In this book, see: Document #55 White Jews' in Science [15 July 1937] pp. 152-157; Document #63

Heinrich Himmler: Letter to Reinhard Heydrich [21 July 1938] pp. 175-176; Document #64 Heinrich Himmler: Letter to Werner Heisenberg

[21 July 1938] pp. 176-177; Document #85 Ludwig Prandtl: Attachment to the letter to Reich Marschal (sic) Hermann Goring [28 April

1941] pp. 261-266; and Document #93 Carl Ramsauer: The Munich Conciliation and Pacification Attempt [20 January 1942] pp. 290-292.
[74] Cassidy, 1992, 390-391. Please note that Cassidy uses the alias Mathias Jules for Johannes Juilfs.
[75] Niels Bohr Archive (http://www.nba.nbi.dk/). His site contains historical background and facsimiles of documents relating to the 1941

Bohr-Heisenberg meeting.

Werner Heisenberg 517

[76] . Heisenberg September 1941 Letter (http://werner-heisenberg.unh.edu/copenhagen.htm)

[77] American Institute for Physics, Center for History of Physics (http://www.aip.org/history/heisenberg/pl4.htm).

[78] Albert Speer, Inside the Third Reich, Macmillan, 1970, pp. 225ff. See also http://www.stanford.edu/~njenkins/cgi-bin/auden/individual.

php?pid=I662&ged=auden-bicknell.ged
[79] Document 98: The Fiihrer's Decree on the Reich Research Council, 9 June 1942, in Hentschel, 1996, 303.
[80] W. Heisenberg Die heohachtburen Grossen in der Theorie der Elementarteilchen. /., Z. Phys. Volume 120, 513-538 (1943), as cited in Mott

and Peierls, 1977, 245.
[81] W. Heisenberg Die heohuchthuren Grossen in der Theorie der Elementarteilchen. II., Z. Phys. Volume 120, 673-702 (1943), as cited in

Mott and Peierls, 1977, 245.
[82] W. Heisenberg Die heohachtburen Grossen in der Theorie der Elementarteilchen. III., Z. Phys. Volume 123, 93-112 (1944), as cited in Mott

and Peierls, 1977, 245.
[83] Jeremy Bernstein Heisenberg in Poland, Am. J. Phys. Volume 72, Number 3, 300-304 (2004). See also Letters to the Editor by Klaus

Gottstein and a reply by Jeremy Bernstein in Am. J. Phys. Volume 72, Number 9, 1 143- 1 145 (2004).
[84] O. Hahn and F. Strassmann Uber den Nachweis und das Verhalten der bei der Bestrahlung des Urans mittels Neutronen entstehenden

Erdalkalimetalle (On the detection and characteristics of the alkaline earth metals formed by irradiation of uranium with neutrons),

Naturwissenschaften Volume 27, Number 1, 11-15 (1939). The authors were identified as being at the Kaiser-Wilhelm-Institut fur Chemie,

Berlin-Dahlem. Received 22 December 1938.
[85] Ruth Lewin Sime Lise Meitner's Escape from Germany, American Journal of Physics Volume 58, Number 3, 263- 267 (1990).
[86] Lise Meitner and O. R. Frisch Disintegration of Uranium by Neutrons: a New Type of Nuclear Reaction, Nature, Volume 143, Number

3615, 239-240 (11 February 1939) (http://www.nature.com/physics/looking-back/meitner/index.html). The paper is dated 16 January

1939. Meitner is identified as being at the Physical Institute, Academy of Sciences, Stockholm. Frisch is identified as being at the Institute of

Theoretical Physics, University of Copenhagen.
[87] O. R. Frisch Physical Evidence for the Division of Heavy Nuclei under Neutron Bombardment, Nature, Volume 143, Number 3616, 276-276

(18 February 1939) (http://dbhs.wvusd.kl2.ca.us/webdocs/Chem-History/Frisch-Fission-1939.html). The paper is dated 17 January

1939. [The experiment for this letter to the editor was conducted on 13 January 1939; see Richard Rhodes The Making of the Atomic Bomb

263 and 268 (Simon and Schuster, 1986).]
[88] In 1944, Hahn received the Nobel Prize for Chemistry for the discovery of nuclear fission. Some historians have documented the history of

the discovery of nuclear fission and believe Meitner should have been awarded the Nobel Prize with Hahn. See the following references: Ruth

Lewin Sime From Exceptional Prominence to Prominent Exception: Lise Meitner at the Kaiser Wilhelm Institute for Chemistry: Ergebnisse 24

(http://www.mpiwg-berlin.mpg.de/KWG/Ergebnisse/Ergebnisse24.pdf) Forschungsprogramm Geschichte der

Kaiser-Wilhelm-Gesellschaft im Nationalsozialismus (2005); Ruth Lewin Sime Lise Meitner: A Life in Physics (University of California,

1997); and Elisabeth Crawford, Ruth Lewin Sime, and Mark Walker A Nobel Tale of Postwar Injustice, Physics Today Volume 50, Issue 9,

26-32 (1997).
[89] Kant, 2002, Reference 8 on p. 3.
[90] Hentschel and Hentschel, 1996, 363-364 and Appendix F; see the entries for Esau, Harteck and Joos. See also the entry for the KWIP in

Appendix A and the entry for the HWA in Appendix B.
[91] Macrakis, 1993, 164-169.

[92] Mehra and Rechenberg, Volume 6, Part 2, 2001, 1010-1011.
[93] Hentschel and Hentschel, 1996, 363-364 and Appendix F; see the entries for Diebner and Dopel. See also the entry for the KWIP in

Appendix A and the entry for the HWA in Appendix B.
[94] Hentschel and Hentschel, 1996; see the entry for the KWIP in Appendix A and the entries for the HWA and the RFR in Appendix B. Also

see p. 372 and footnote #50 on p. 372.
[95] Walker, 1993, 49-53.
[96] Walker, 1993, 52-53.
[97] Kant, 2002, 19.
[98] Deutsches Museum (http://www.deutsches-museum.de/archiv/archiv-online/geheimdokumente/forschungszentren/

wien-heidelberg-strassburg/taetigkeitsbericht-pi-wien/) - Tdtigkeitsbericht des II. Physikalischen Instituts der Wiener Universitat, 1945
[99] Walker, 1993, 19 and 94-95.
[100] Walker, 1993, 52 and Reference #40 on p. 262.
[101] Walker, 1993, 208.

[102] Hentschel and Hentschel, 1996, Appendix F; see the entry for Schumann. Also see footnote #1 on p. 207.
[103] Goudsmit, Samuel with an introduction by R. V. Jones Alsos (Toamsh, 1986).
[104] Pash, Boris T. The Alsos Mission (Award, 1969).
[105] Cassidy, Uncertainty, 1992,491-500.
[106] Naimark, 1995, 208-209.
[107] Bernstein, 2001,49-52.

[108] Walker, 1993, 268-274 and Reference #40 on p. 262.
[109] Bernstein, 2001, 50 and 363-365.
[110] Cassidy, Uncertainty, 1992,491-510.

Werner Heisenberg 518

[111] Bernstein, 2001,60.

[112] Pash, Boris T. The Alsos Mission (Award, 1969) pp. 219-241.

[113] Charles Franck Operation Epsilon: The Farm Hall Transcripts (University of California Press, 1993)

[114] Jeremy Bernstein Hitler's Uranium Club: The Secret Recording's at Farm Hall (Copernicus, 2001) ISBN 0-387-95089-3.

[115] Bernstein, 2001, xvii-xix.

[116] Walker, 1993, 184-185.

[117] Oleynikov, 2000, 14.

[118] W. Heisenberg Zur Theorie tier Supraleitung, Forsch. Fortschr. Volumes 21/23, 243-244 (1947); Z. Naturf. Volume 2a, 185-201 (1947),

cited in Mott and Peierls, 1977, 245.
[119] W. Heisenberg Das elektrodynamische Verhalten der Supraleiter, Z. Naturf. Volume 3a, 65-75 (1948), cited in Mott and Peierls, 1977,

245.
[120] M. von Laue and W. Heisenberg Das Barlowsche Rad aus supraleitendem Material, Z. Phys. Volume 124, 514-518 (1948), cited in Mott

and Peierls, 1977, 245.
[121] Mott and Peierls, 1977,238-239.

[122] W. Heisenberg Zur statistischen Theorie der Tuhulenz, Z. Phys. Volume 124, 628-657 (1948), as cited in Mott and Peierls, 1977, 245.
[123] W. Heisenberg On the theory of statistical and isotropic turbulence, Proc. R. Soc. London A Volume 195, 402-406 (1948), as cited in Mott

and Peierls, 1977, 245.
[124] W. Heisenberg Bemerkungen um Turbulenzproblem, Z. Naturf. Volume 3a, 434-437 (1948), as cited in Mott and Peierls, 1977, 245.
[125] W. Heisenberg On the stability of laminar flow , Proc. International Congress Mathematicians Volume II, 292-296 (1950), as cited in Mott

and Peierls, 1977, 245.
[126] W. Heisenberg Production of mesons showers, Nature, Land. Volume 164, 65-67 (1949), as cited in Mott and Peierls, 1977, 245.
[127] W. Heisenberg Die Erzeugung von Mesonen in Vielfachprozessen, Nuovo Cimento Volume 6 (Supplement), 493-497 (1949) as cited in

Mott and Peierls, 1977, 245.
[128] W. Heisenberg Uber die Entstehung von Mesonen in Vielfachprozessen, Z. Phys. Volume 126, 569-582 (1949), as cited in Mott and

Peierls, 1977, 245.
[129] W. Heisenberg Bermerkungen zur Theorie der Vielfacherzeugung von Mesonen, Die Naturwissenschaften Volume 39, 69 (1952), as cited

in Mott and Peierls, 1977, 246.
[130] W. Heisenberg Mesonenerzeugung als Stosswellenproblem, Z. Phys. Volume 133, 65-79 (1952), as cited in Mott and Peierls, 1977, 246.
[131] W, Heisenberg The production of mesons in very high energy collisions, Nuovo Cimento Volume 12, Supplement, 96-103 (1955), as cited

in Mott and Peierls, 1977, 246.
[132] Mott and Peierls, 1977, 238.

[133] Hentschel, 1996, Appendix A; see the entries for DFG and NG.

[134] J. L. Heilbron The Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science (Harvard, 2000) pp. 90-92.
[135] Cassidy, Uncertainty, 1992, 262.
[136] Horst Kant Werner Heisenberg and the German Uranium Project / Otto Hahn and the Declarations ofMainau and Gottingen, Preprint 203

(Max-Planck Institut filr Wissenschaftsgeschichte, 2002 (http://www.mpiwg-berlin.mpg.de/Preprints/P203.PDF)).
[137] Declaration of the German Nuclear Physicists ArmsControl.de (http://www.armscontrol.de/dokumente/goettingen-eng.pdf).
[138] Werner Heisenberg Development of concepts in the history of quantum theory, American Journal of Physics Volume 43, Number 5,

389-394. The substance of this article was presented by Heisenberg in a lecture at Harvard University.
[139] Chapter 16 "Scientific and Religious Truth" in Across the Frontiers, 1974, Harper & Row, p. 213-229
[140] Cassidy, Uncertainty, 1992, 372 and Appendix A.

[141] David Cassidy and the American Institute of Physics, The Difficult Years (http://www.aip.org/history/heisenberg/plO.htm).
[142] http://www.adherents.com/people/ph/Werner_Heisenberg.html
[143] Cassidy, Uncertainty, 1992, 262, 545.
[144] Cassidy, Uncertainty, 1992, 545.

[145] Hentschel and Hentschel, 1996, Appendix E; see the entry for Kernphysikalische Forschungsberichte.
[146] Walker, 1993, 268-274.
[147] Prdparat 38 was the cover name for uranium oxide; see Deutsches Museum (http://www.deutsches-museum.de/archiv/archiv-online/

geheimdokumente/forschungszentren/leipzig/schichtenanordnung-h2o/).
[148] http://www.aip.org/history/heisenberg/bibliography/1920-29.htm
[149] http://www.aip.org/history/heisenberg/bibliography/pdf/1920-29.pdf
[150] http://arxiv.org/abs/physics/0605038vl
[151] http://www.archive.org/details/physicsandphilos010613mbp

Werner Heisenberg 519

References

Bernstein, Jeremy and David Cassidy Bomb Apologetics: Farm Hall, August 1945, Physics Today Volume 48,

Issue 8, Part I, 32-36 (1995)

Bernstein, Jeremy Hitler's Uranium Club: The Secret Recording's at Farm Hall (Copernicus, 2001) ISBN

0-387-95089-3

Bernstein, Jeremy Heisenberg and the critical mass, Am. J. Phys. Volume 70, Number 9, 911-916 (2002)

Bernstein, Jeremy Heisenberg in Poland, Am. J. Phys. Volume 72, Number 3, 300-304 (2004). See also Letters to

the Editor by Klaus Gottstein and a reply by Jeremy Bernstein in Am. J. Phys. Volume 72, Number 9, 1 143-1 145

(2004).

Bernstein, Jeremy Max Born and the Quantum Theory, Am. J. Phys. 73 (1 1) 999-1008 (2005). Department of

Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030. Received 14 April 2005; accepted 29 July

2005.

Bethe, Hans A. The German Uranium Project, Physics Today Volume 53, Issue 7, 34-36

Beyerchen, Alan D. Scientists Under Hitler: Politics and the Physics Community in the Third Reich (Yale, 1977)

ISBN 0-300-01830-4

Cassidy, David C. Heisenberg, German Science, and the Third Reich, Social Research Volume 59, Number 3,

643-661 (1992)

Cassidy, David C. Uncertainty: The Life and Science of Werner Heisenberg (Freeman, 1992)

Cassidy, David C. A Historical Perspective on Copenhagen, Physics Today Volume 53, Issue 7, 28 (2000). See

also Heisenberg 's Message to Bohr: Who Knows, Physics Today Volume 54, Issue 4, 14ff (2001), individual

letters by Klaus Gottstein, Harry J. Lipkin, Donald C. Sachs, and David C. Cassidy.

Chevalley, Catherine Werner Heisenberg: Philosophie le Manuscrit de 1942 (Editions du Seuil, 1998)

Eckert, Michael Primacy doomed to failure: Heisenberg 's role as scientific adviser for nuclear policy in the FRG,

Historical Studies in the Physical and Biological Sciences Volume 21, Number 1, 29 — 58 (1990)

Eckert, Michael Werner Heisenberg: controversial scientist physicsweb.org (2001) (http://physicsworld.com/

cws/article/print/3462)

Goudsmit, Samuel with an introduction by R. V. Jones Alsos (Toamsh, 1986)

Fedak, William A. and Jeffrey J. Prentis The 1925 Born and Jordan paper "On quantum mechanics", American

Journal of Physics Volume 77, Number 2, pp. 128 - 139 (2009)

Greenspan, Nancy Thorndike "The End of the Certain World: The Life and Science of Max Born" (Basic Books,

2005) ISBN 0-7382-0693-8. Also published in Germany: Max Born — Baumeister der Quantenwelt. Eine

Biographie (Spektrum Akademischer Verlag, 2005), ISBN 3-8274- 1640-X.

Heisenberg, Werner Nobel Prize Presentation Speech, Nobelprize.org (1933) (http://nobelprize.org/

nobel_prizes/physics/laureates/1932/press.html)

• Werner Heisenberg Biography (http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-bio.

html), Nobel Prize in Physics 1932 Nobelprize.org
Heisenberg, Elisabeth Inner Exile: Recollections of a Life with Heisenberg (Birkhauser, 1984)
Heisenberg, Werner Physics and Beyond: Encounters and Conversations (Harper & Row, 1971)
Heisenberg, Werner Die theoretischen Grundlagen fur die Energiegewinnung aus der Uranspaltung, Zeitschrift
fiir die gesamte Natruwiessenschaft, Volume 9, 201-212 (1943). See also the annotated English translation:
Document 95. Werner Heisenberg. The Theoretical Basis for the Generation of Energy from Uranium Fission [26
February 1942] in Hentschel, Klaus (editor) and Ann M. Hentschel (editorial assistant and translator) Physics and
National Socialism: An Anthology of Primary Sources (Birkhauser, 1996) 294-301.

Heisenberg, Werner Research in Germany on the Technical Applications of Atomic Energy, Nature Volume 160,
Number 4059, 211-215 (16 August 1947). See also the annotated English translation: Document 115. Werner
Heisenberg: Research in Germany on the Technical Application of Atomic Energy [16 August 1947] in
Hentschel, Klaus (editor) and Ann M. Hentschel (editorial assistant and translator) Physics and National

Werner Heisenberg 520

Socialism: An Anthology of Primary Sources (Birkhauser, 1996) 361-379.

• Heisenberg, Werner, introduction by David Cassidy, translation by William Sweet A Lecture on Bomb Physics:
February 1942, Physics Today Volume 48, Issue 8, Part I, 27-30 (1995)

• Hentschel, Klaus (editor) and Ann M. Hentschel (editorial assistant and translator) Physics and National
Socialism: An Anthology of Primary Sources (Birkhauser, 1996) ISBN 0-8176-5312-0. [This book is a collection
of 121 primary German documents relating to physics under National Socialism. The documents have been
translated and annotated, and there is a lengthy introduction to put them into perspective.]

• Hentschel, Klaus The Metal Aftermath: The Mentality of German Physicists 1945-1949 (Oxford, 2007)

• Hoffmann, Dieter Between Autonomy and Accommodation: The German Physical Society during the Third Reich,
Physics in Perspective 7(3) 293-329 (2005)

• Jammer, Max The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966)

• Junk, Robert Brighter Than a Thousand Suns: A personal history of the atomic scientists (Harcourt, Brace, 1958)

• Kant, Horst Werner Heisenberg and the German Uranium Project / Otto Hahn and the Declarations ofMainau
and Gottingen, Preprint 203 (Max-Planck Institut fur Wissenschaftsgeschichte, 2002 (http://www.mpiwg-berlin.
mpg . de/Preprints/P203 .PDF))

• Landsman, N. P. Getting even with Heisenberg, Studies in History and Philosophy of Modern Physics Volume 33,
297-325 (2002)

• MacKinnon, Edward, "Heisenberg, Models, and the Rise of Quantum Mechanics", Historical Studies in the
Physical Sciences, Volume 8, 137-188 (1977)

• Macrakis, Kristie Surviving the Swastika: Scientific Research in Nazi Germany (Oxford, 1993)

• Mehra, Jagdish, and Helmut Rechenberg The Historical Development of Quantum Theory. Volume 1 Part 2 The
Quantum Theory of Planck, Einstein, Bohr and Sommerfeld 1900—1925: Its Foundation and the Rise of Its
Difficulties. (Springer, 2001) ISBN 0-387-95 175-X

• Mehra, Jagdish and Helmut Rechenberg The Historical Development of Quantum Theory. Volume 3. The
Formulation of Matrix Mechanics and Its Modifications 1925—1926. (Springer, 2001) ISBN 0-387-95177-6

• Mehra, Jagdish and Helmut Rechenberg The Historical Development of Quantum Theory. Volume 6. The
Completion of Quantum Mechanics 1926-1941. Part 2. The Conceptual Completion and Extension of Quantum
Mechanics 1932-1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942-1999. (Springer,
2001) ISBN 978-0-387-95086-0

• Mott, N. and R. Peierls Werner Heisenberg, Biographical Memoirs of Fellows of the Royal Society Volume 23,
213-251 (1977)

• Norman M. Naimark The Russians in Germany: A History of the Soviet Zone of Occupation, 1945-1949
(Belkanp, 1995)

• Oleynikov, Pavel V. German Scientists in the Soviet Atomic Project, The Nonproliferation Review Volume 7,
Number 2, 1—30 (2000) (http://cns.miis.edu/pubs/npr/vol07/72/72pavel.pdf). The author has been a group
leader at the Institute of Technical Physics of the Russian Federal Nuclear Center in Snezhinsk (Chelyabinsk-70).

• Pash, Boris T. The Alsos Mission (Award, 1969)

• Powers, Thomas Heisenberg' s War: The Secret History of the German Bomb (Knopf, 1993)

• Rose, Paul Lawrence, Heisenberg and the Nazi Atomic Bomb Project: A Study in German Culture (California,
1998). For a critical review of this book, please see: Landsman, N. P. Getting even with Heisenberg, Studies in
History and Philosophy of Modern Physics Volume 33, 297-325 (2002).

• Todorv, Ivan Werner Heisenberg ( 2003 (http://arxiv.org/abs/physics/0503235))

• van der Waerden, B. L., editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1

• Walker, Mark Heisenberg, Goudsmit and the German Atomic Bomb, Physics Today Volume 43, Issue 1, 52-60
(1990)

• Walker, Mark Physics and propaganda: Werner Heisenberg 's foreign lectures under National Socialism,
Historical Studies in the Physical Sciences Volume 22, 339-389 (1992)

Werner Heisenberg 521

• Walker, Mark German National Socialism and the Quest for Nuclear Power 1939—1949 (Cambridge, 1993) ISBN
0-521-43804-7

• Walker, Mark Eine Waff ens chmiede? Kernwaffen- und Reaktorforschung am Kaiser-Wilhelm-Institut fur Physik,
Forschungsprogramm "Geschichte der Kaiser-Wilhelm-Gesellschaft im Nationalsozialismus" Ergebnisse 26
(http://www.mpiwg-berlin.mpg.de/KWG/Ergebnisse/Ergebnisse26.pdf) (2005)

Further reading

Born, Max The statistical interpretation of quantum mechanics. Nobel Lecture (http://nobelprize.org/

nobel_prizes/physics/laureates/1954/born-lecture.pdf) — 11 December 1954.

Cassidy, David C. Werner Heisenberg : A Bibliography of His Writings, Second, Expanded Edition (Whittier,

2001)

Dorries, Matthias Michael Frayn's 'Copenhagen ' in Debate: Historical Essays and Documents on the 1941

Meeting Between Niels Bohr and Werner Heisenberg (University of California, 2005)

Fischer, Ernst P. Werner Heisenberg: Das selbstvergessene Genie (Piper, 2002)

Heisenberg, Werner "A Scientist's case for the Classics" (Harper's Magazine, May 1958, p. 25-29)

Heisenberg, Werner Across the Frontiers (Harper & Row, 1974)

Kleint, Christian and Gerald Wiemer Werner Heisenberg im Spiegel seiner Leipziger Schiller und Kollegen

(Leipziger Universitatsverlag, 2005)

PapenfuB, Dietrich, Dieter Lust, and Wolfgang P. Schleich 100 Years Werner Heisenberg: Works and Impact

(Wiley- VCH, 2002)

Rechenberg, Helmut und Gerald Wiemers Werner Heisenberg (1901—1976), Schritte in die neue Physik

(Sax-Verlag Beucha, 2001)

Schiemann, Gregor Werner Heisenberg (C.H. Beck, 2008)

von Weizsacker, Carl Friedrich and B artel Leendert van der Waerden Werner Heisenberg (Hanser, Carl GmbH,

1977)

Rhodes, Richard The Making of the Atomic Bomb (Simon and Schuster, 1986)

Walker, Mark National Socialism and German Physics, Journal of Contemporary Physics Volume 24, 63-89

(1989)

Walker, Mark Nazi Science: Myth, Truth, and the German Atomic Bomb (Perseus, 1995)

Walker, Mark German Work on Nuclear Weapons, Historia Scientiarum; International Journal for the History of

Science Society of Japan, Volume 14, Number 3, 164-181 (2005)

External links

• Annotated Bibliography for Werner Heisenberg from the Alsos Digital Library for Nuclear Issues (http://alsos.
wlu.edu/qsearch. aspx?browse=people/Heisenberg,+Werner)

• MacTutor Biography: Werner Karl Heisenberg (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/
Heisenberg. html)

• Oral history interview transcript with Werner Heisenberg, 30 November 1962, American Institute of Physics,
Niels Bohr Library & Archives (http://www.aip.org/history/ohilist/4661_l.html)

• Oral history interview transcript with Werner Heisenberg, 16 June 1970, American Institute of Physics, Niels
Bohr Library & Archives (http://www.aip.org/history/ohilist/5027.html)

• Key Participants: Werner Heisenberg (http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/
people/heisenberg.html) - Linus Pauling and the Nature of the Chemical Bond: A Documentary History

• Nobelprize.org biography (http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-bio.html)

Albert Einstein

522

Albert Einstein

Albert Einstein in 1921

Born
Died

Residence
Citizenship

Alma mater

Known for

Spouse

Awards

14 March 1879Ulm, Kingdom of Wurttemberg, German Empire
18 April 1955 (aged 76)Princeton, New Jersey, United States

Germany, Italy, Switzerland, United States

Wurttemberg/Germany (until 1896)
Stateless (1896-1901)
Switzerland (from 1901)
Austria (1911-12)
Germany (1914-33)

United States (from 1940)

ETH Zurich
University of Zurich

[1]

General relativity and special relativity
Photoelectric effect
Mass-energy equivalence
Quantification of the Brownian motion
Einstein field equations
Bose— Einstein statistics
Unified Field Theory

Mileva Marie (1903-1919)

Elsa Lowenthal, nee Einstein, (1919—1936)

Nobel Prize in Physics (1921)
Copley Medal (1925)
Max Planck Medal (1929)
Time Person of the Century

Signature

Albert Einstein ( 4) /'aelbert'alnstaln/; German: ['albet 'alnjtaln] ( 49 listen); 14 March 1879 - 18 April 1955) was a

German-born theoretical physicist who discovered the theory of general relativity, effecting a revolution in physics.

T21
For this achievement, Einstein is often regarded as the father of modern physics. He received the 1921 Nobel Prize

in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric

Albert Einstein

523

effect

, [3]

Near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the
laws of classical mechanics with the laws of the electromagnetic field. This led to the development of his special
theory of relativity. He realized, however, that the principle of relativity could also be extended to gravitational
fields, and with his subsequent theory of gravitation in 1916, he published a paper on the general theory of relativity.
He continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of
particle theory and the motion of molecules. He also investigated the thermal properties of light which laid the
foundation of the photon theory of light. In 1917, Einstein applied the general theory of relativity to model the

structure of the universe as a whole

[4]

He was visiting the United States when Hitler came to power in 1933, and did not go back to Germany, where he had
been a professor at the Berlin Academy of Sciences. He settled in the U.S., becoming a citizen in 1940. On the eve of
World War II, he helped alert President Franklin D. Roosevelt that Germany might be developing an atomic weapon,
and recommended that the U.S. begin similar research. Later, together with Bertrand Russell, Einstein signed the
Russell— Einstein Manifesto, which highlighted the danger of nuclear weapons. Einstein taught physics at the
Institute for Advanced Study at Princeton, New Jersey, until his death in 1955.

Einstein published more than 300 scientific papers along with over 150 non-scientific works
intelligence and originality have made the word "Einstein" synonymous with genius.

[4] [5]

His great

Biography

Early life and education

Albert Einstein was born in Ulm, in the Kingdom of Wurttemberg in the German

T71
Empire on 14 March 1879. His father was Hermann Einstein, a salesman and

engineer. His mother was Pauline Einstein (nee Koch). In 1880, the family

moved to Munich, where his father and his uncle founded Elektrotechnische

Fabrik J. Einstein & Cie, a company that manufactured electrical equipment

based on direct current

[71

The Einsteins were non-observant Jews. Albert attended a Catholic elementary
school from the age of five for three years. Later, at the age of eight, Einstein
was transferred to the Luitpold Gymnasium where he received advanced primary

ro]

and secondary school education till he left Germany seven years later.
Although it has been thought that Einstein had early speech difficulties, this is
disputed by the Albert Einstein Archives, and he excelled at the first school that
he attended

[91

Einstein at the age of 4

His father once showed him a pocket compass; Einstein realized that there must

be something causing the needle to move, despite the apparent "empty space". As he grew, Einstein built models
and mechanical devices for fun and began to show a talent for mathematics. In 1889, Max Talmud (later changed
to Max Talmey) introduced the ten-year old Einstein to key texts in science, mathematics and philosophy, including
Immanuel Kant's Critique of Pure Reason and Euclid's Elements (which Einstein called the "holy little geometry
book"). Talmud was a poor Jewish medical student from Poland. The Jewish community arranged for Talmud to
take meals with the Einsteins each week on Thursdays for six years. During this time Talmud wholeheartedly guided

Einstein through many secular educational interests

[12] [13]

Albert Einstein

524

In 1894, his father's company failed: direct current (DC) lost the War of Currents
to alternating current (AC). In search of business, the Einstein family moved to
Italy, first to Milan and then, a few months later, to Pavia. When the family
moved to Pavia, Einstein stayed in Munich to finish his studies at the Luitpold
Gymnasium. His father intended for him to pursue electrical engineering, but
Einstein clashed with authorities and resented the school's regimen and teaching
method. He later wrote that the spirit of learning and creative thought were lost

in strict rote learning. In the spring of 1895, he withdrew to join his family in

171
Pavia, convincing the school to let him go by using a doctor's note. During this

time, Einstein wrote his first scientific work, "The Investigation of the State of

ri4i

Aether in Magnetic Fields".

Albert Einstein in 1893 (age 14)

Einstein applied directly to the Eidgenossische Polytechnische Schule (ETH) in Zurich, Switzerland. Lacking the
requisite Matura certificate, he took an entrance examination, which he failed, although he got exceptional marks in

mathematics and physics. The Einsteins sent Albert to Aarau, in northern Switzerland to finish secondary

171
school. While lodging with the family of Professor Jost Winteler, he fell in love with Winteler's daughter, Marie.

(His sister Maja later married the Wintelers' son, Paul.) In Aarau, Einstein studied Maxwell's electromagnetic

theory. At age 17, he graduated, and, with his father's approval, renounced his citizenship in the German Kingdom of

Wurttemberg to avoid military service, and in 1896 he enrolled in the four year mathematics and physics teaching

diploma program at the Polytechnic in Zurich. Marie Winteler moved to Olsberg, Switzerland for a teaching post.

Einstein's future wife, Mileva Marie, also enrolled at the Polytechnic that same year, the only woman among the six
students in the mathematics and physics section of the teaching diploma course. Over the next few years, Einstein
and Marie's friendship developed into romance, and they read books together on extra-curricular physics in which

Einstein was taking an increasing interest. In 1900 Einstein was awarded the Zurich Polytechnic teaching diploma,

1171
but Marie failed the examination with a poor grade in the mathematics component, theory of functions. There

have been claims that Marie collaborated with Einstein on his celebrated 1905 papers, but historians of

[201 r211 T221 T231

physics who have studied the issue find no evidence that she made any substantive contributions.

Marriages and children

In early 1902, Einstein and Mileva Marie had a daughter they named Lieserl in their correspondence, who was born

1241 T251

in Novi Sad where Marie's parents lived. Her full name is not known, and her fate is uncertain after 1903.

Einstein and Marie married in January 1903. In May 1904, the couple's first son, Hans Albert Einstein, was born in
Bern, Switzerland. Their second son, Eduard, was born in Zurich in July 1910. In 1914, Einstein moved to Berlin,
while his wife remained in Zurich with their sons. Marie and Einstein divorced on 14 February 1919, having lived
apart for five years.

Einstein married Elsa Lowenthal (nee Einstein) on 2 June 1919, after having had a relationship with her since 1912.
She was his first cousin maternally and his second cousin paternally. In 1933, they emigrated permanently to the
United States. In 1935, Elsa Einstein was diagnosed with heart and kidney problems and died in December 1936.

Albert Einstein

525

Patent office

Left to right: Conrad Habicht, Maurice Solovine
and Einstein, who founded the Olympia Academy

After graduating, Einstein spent almost two frustrating years searching
for a teaching post, but a former classmate's father helped him secure a
job in Bern, at the Federal Office for Intellectual Property, the patent

1271

office, as an assistant examiner. He evaluated patent applications
for electromagnetic devices. In 1903, Einstein's position at the Swiss
Patent Office became permanent, although he was passed over for
promotion until he "fully mastered machine technology

„ [28]

Much of his work at the patent office related to questions about
transmission of electric signals and electrical-mechanical
synchronization of time, two technical problems that show up
conspicuously in the thought experiments that eventually led Einstein
to his radical conclusions about the nature of light and the fundamental
connection between space and time.

With a few friends he met in Bern, Einstein started a small discussion
group, self-mockingly named "The Olympia Academy", which met
regularly to discuss science and philosophy. Their readings included
the works of Henri Poincare, Ernst Mach, and David Hume, which
influenced his scientific and philosophical outlook.

Academic career

Einstein's official 1921 portrait after receiving
the Nobel Prize in Physics.

In 1901, Einstein had a paper on the capillary forces of a straw
published in the prestigious Annalen der Physik. On 30 April 1905,
he completed his thesis, with Alfred Kleiner, Professor of Experimental
Physics, serving as pro-forma advisor. Einstein was awarded a PhD by

the University of Zurich. His dissertation was entitled "A New

T311
Determination of Molecular Dimensions". That same year, which has

been called Einstein's annus mirabilis or "miracle year", he published

four groundbreaking papers, on the photoelectric effect, Brownian

motion, special relativity, and the equivalence of matter and energy,

which were to bring him to the notice of the academic world.

By 1908, he was recognized as a leading scientist, and he was appointed

lecturer at the University of Bern. The following year, he quit the patent

132]
office and the lectureship to take the position of physics docent at the

University of Zurich. He became a full professor at Karl-Ferdinand

University in Prague in 1911. In 1914, he returned to Germany after

being appointed director of the Kaiser Wilhelm Institute for Physics

T331
(1914—1932) and a professor at the Humboldt University of Berlin,

Albert Einstein 526

although with a special clause in his contract that freed him from most teaching obligations. He became a member of
the Prussian Academy of Sciences. In 1916, Einstein was appointed president of the German Physical Society
(1916-1918). [34] [35]

In 1911, he had calculated that, based on his new theory of general relativity, light from another star would be bent
by the Sun's gravity. That prediction was claimed confirmed by observations made by a British expedition led by Sir
Arthur Eddington during the solar eclipse of May 29, 1919. International media reports of this made Einstein world
famous. On 7 November 1919, the leading British newspaper The Times printed a banner headline that read:
"Revolution in Science — New Theory of the Universe — Newtonian Ideas Overthrown". (Much later, questions
were raised whether the measurements were accurate enough to support Einstein's theory.)

In 1921, Einstein was awarded the Nobel Prize in Physics. Because relativity was still considered somewhat
controversial, it was officially bestowed for his explanation of the photoelectric effect. He also received the Copley
Medal from the Royal Society in 1925.

Travels abroad

Einstein visited New York City for the first time on 2 April 1921, where he received an official welcome by the

Mayor, followed by three weeks of lectures and receptions. He went on to deliver several lectures at Columbia

University and Princeton University, and in Washington he accompanied representatives of the National Academy of

Science on a visit to the White House. On his return to Europe he was the guest of the British statesman and

philosopher Viscount Haldane in London, where he met several renowned scientific, intellectual and political

[371
figures, and delivered a lecture at Kings College.

In 1922, he traveled throughout Asia and later to Palestine, as part of a six-month excursion and speaking tour. His
travels included Singapore, Ceylon, and Japan, where he gave a series of lectures to thousands of Japanese. His first
lecture in Tokyo lasted four hours, after which he met the emperor and empress at the Imperial Palace where

hoi -307

thousands came to watch. Einstein later gave his impressions of the Japanese in a letter to his sons: ' "Of all the

r^Q]

people I have met, I like the Japanese most, as they are modest, intelligent, considerate, and have a feel for art."

:308

On his return voyage, he also visited Palestine for 12 days in what would become his only visit to that region. "He
was greeted with great British pomp, as if he were a head of state rather than a theoretical physicist", writes Isaacson.
This included a cannon salute upon his arrival at the residence of the British high commissioner, Sir Herbert Samuel.
During one reception given to him, the building was "stormed by throngs who wanted to hear him". In Einstein's talk
to the audience, he expressed his happiness over the event:

I consider this the greatest day of my life. Before, I have always found something to regret in the Jewish
soul, and that is the forgetfulness of its own people. Today, I have been made happy by the sight of the

Jewish people learning to recognize themselves and to make themselves recognized as a force in the

, , [39] :308
world.

Albert Einstein

527

Emigration from Germany

Cartoon of Einstein, who has shed his "Pacifism"

wings, standing next to a pillar labeled "World

Peace." He is rolling up his sleeves and holding a

sword labeled "Preparedness." (circa 1933)

In 1933, Einstein decided to emigrate to the United States due to the

rise to power of the Nazis under Germany's new chancellor, Adolf

Hitler. While visiting American universities in April, 1933, he

learned that the new German government had passed a law barring

Jews from holding any official positions, including teaching at

universities. A month later, the Nazi book burnings occurred, with

Einstein's works being among those burnt, and Nazi propaganda

minister Joseph Goebbels proclaimed, "Jewish intellectualism is

[39]
dead." Einstein also learned that his name was on a list of

assassination targets, with a "$5,000 bounty on his head". One German

magazine included him in a list of enemies of the German regime with

the phrase, "not yet hanged

.. [39]

Einstein was undertaking his third two-month visiting professorship at
the California Institute of Technology when Hitler came to power in
Germany. On his return to Europe in March 1933 he resided in

Belgium for some months, before temporarily moving to England

[42]

[41]

He took up a position at the Institute for Advanced Study at Princeton, New Jersey, an affiliation that lasted until
his death in 1955. There, he tried to develop a unified field theory and to refute the accepted interpretation of
quantum physics, both unsuccessfully. He and Kurt Godel, another Institute member, became close friends. They
would take long walks together discussing their work. His last assistant was Bruria Kaufman, who later became a
renowned physicist.

Other scientists also fled to America. Among them were Nobel laureates and professors of theoretical physics. With
so many other Jewish scientists now forced by circumstances to live in America, often working side by side, Einstein
wrote to a friend, "For me the most beautiful thing is to be in contact with a few fine Jews — a few millennia of a
civilized past do mean something after all." In another letter he writes, "In my whole life I have never felt so Jewish

as now

,,[39]

World War II and the Manhattan Project

In 1939, a group of Hungarian scientists that included Hungarian emigre physicist Leo Szilard attempted to alert

[43]

Washington of ongoing Nazi atomic bomb research. The group's warnings were discounted.

In the summer of 1939, a few months before the beginning of World War II in Europe, Einstein was persuaded to
lend his prestige by writing a letter, with Leo Szilard, to President Franklin D. Roosevelt, in order to alert him of the
possibility that Nazi Germany might be developing an atomic bomb. At the same time, the letter recommended that
the U.S. government should pay attention to and become directly involved with uranium research, and associated
chain reaction research. Einstein and Szilard, along with other refugees such as Edward Teller and Eugene Wigner,
"regarded it as their responsibility to alert Americans to the possibility that German scientists might win the race to
build an atomic bomb, and to warn that Hitler would be more than willing to resort to such a weapon."

The letter is believed to be "arguably the key stimulus for the U.S. adoption of serious investigations into nuclear

[45]

weapons on the eve of the U.S. entry into World War II". President Roosevelt could not take the risk of allowing
Hitler to possess atomic bombs first. As a result of Einstein's letter and his meetings with Roosevelt, the U.S. entered
the "race" to develop the bomb, drawing on its "immense material, financial, and scientific resources" to initiate the
Manhattan Project. It became the only country to develop an atomic bomb during World War n.

For Einstein, "war was a disease . . . [and] he called for resistance to war." But in 1933, after Hitler assumed full
power in Germany, "he renounced pacifism altogether ... In fact, he urged the Western powers to prepare

Albert Einstein

528

themselves against another German onslaught." ' In 1954, a year before his death, Einstein said to his old
friend, Linus Pauling, "I made one great mistake in my life — when I signed the letter to President Roosevelt
recommending that atom bombs be made; but there was some justification — the danger that the Germans would

make them.

,,[47]

U.S. citizenship

Einstein became an American citizen in 1940. Not long after settling
into his career at Princeton, he expressed his appreciation of the
"meritocracy" in American culture when compared to Europe.
According to Isaacson, he recognized the "right of individuals to say
and think what they pleased", without social barriers, and as result, the
individual was "encouraged" to be more creative, a trait he valued from
his own early education. Einstein writes:

What makes the new arrival devoted to this country is the
democratic trait among the people. No one humbles
himself before another person or class. . . American youth
has the good fortune not to have its outlook troubled by

outworn traditions

[39] :432

As a member of the National Association for the Advancement of
Colored People NAACP at Princeton who campaigned for the civil
rights of African Americans, Einstein corresponded with civil rights
activist W. E. B. Du Bois, and in 1946 Einstein called racism
America's "worst disease". He later stated, "Race prejudice has
unfortunately become an American tradition which is uncritically
handed down from one generation to the next. The only remedies are
enlightenment and education

„ [49]

Einstein with David Ben Gurion,
1951

After the death of Israel's first president, Chaim Weizmann, in

November 1952, Prime Minister David Ben-Gurion offered Einstein

the position of President of Israel, a mostly ceremonial post. The offer was presented by Israel's ambassador in

Washington, Abba Eban, who explained that the offer "embodies the deepest respect which the Jewish people can

T3S1 - 522

repose in any of its sons". ' However, Einstein declined, and wrote in his response that he was "deeply moved",
and "at once saddened and ashamed" that he could not accept it:

All my life I have dealt with objective matters, hence I lack both the natural aptitude and the experience
to deal properly with people and to exercise official function. I am the more distressed over these
circumstances because my relationship with the Jewish people became my strongest human tie once I
achieved complete clarity about our precarious position among the nations of the world.

Albert Einstein

529

Death

On April 17, 1955, Albert Einstein experienced internal bleeding
caused by the rupture of an abdominal aortic aneurysm, which had
previously been reinforced surgically by Dr. Rudolph Nissen in
1948. He took the draft of a speech he was preparing for a
television appearance commemorating the State of Israel's seventh
anniversary with him to the hospital, but he did not live long enough to
complete it. Einstein refused surgery, saying: "I want to go when I
want. It is tasteless to prolong life artificially. I have done my share, it
is time to go. I will do it elegantly." He died in Princeton Hospital
early the next morning at the age of 76, having continued to work until
near the end.

1 New York World Telegram

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The New York World-Telegram announces
Einstein's death on April 18, 1955.

Einstein's remains were cremated and his ashes were scattered at an undisclosed location. During the autopsy,

the pathologist of Princeton Hospital, Thomas Stoltz Harvey, removed Einstein's brain for preservation, without the

permission of his family, in hope that the neuroscience of the future would be able to discover what made Einstein so

1571
intelligent. In his lecture at Einstein's memorial, nuclear physicist Robert Oppenheimer summarized his

impression of him as a person:

"He was almost wholly without sophistication and wholly without worldliness
with him a wonderful purity at once childlike and profoundly stubborn."

There was always

Scientific career

[5] [7]

Throughout his life, Einstein published hundreds of books and articles.

In addition to the work he did by himself he also collaborated with other

scientists on additional projects including the Bose— Einstein statistics, the

T5R1

Einstein refrigerator and others.

Annus Mirabilis papers

The Annus Mirabilis papers are four articles pertaining to the photoelectric

effect, Brownian motion, the special theory of relativity, and E = mc that
Albert Einstein published in the Annalen der Physik scientific journal in 1905.
These four works contributed substantially to the foundation of modern
physics and changed views on space, time, and matter.

Thermodynamic fluctuations and statistical physics

[59]
Albert Einstein's first paper submitted in 1900 to Annalen der Physik was on capillary attraction. It was published

in 1901 titled Folgerungen aus den Capillaritdtserscheinungen, which was translated as "Conclusions from the

capillarity phenomena". Two papers he published in 1902-1903 (thermodynamics) attempted to interpret phenomena

from a statistical atomic point of view. These

Albert Einstein

530

papers were the foundation for the 1905 paper on Brownian
motion. These published calculations (1905) showed that
Brownian movement can be construed as firm evidence that
molecules exist.

His research in 1903 and 1904 was mainly concerned with the

[59]
effect of finite atomic size on diffusion phenomena. Einstein's

theory of Brownian motion was the first paper in the field of

statistical physics. It is the most frequently cited, of the Annus

Mirabilis papers.

>© © © ©

The photoelectric effect. Incoming photons on the left

strike a metal plate (bottom), and eject electrons,

depicted as flying off to the right.

General principles postulated by Einstein

He articulated the principle of relativity. This was understood by Hermann Minkowski to be a generalization of
rotational invariance from space to space-time. Other principles postulated by Einstein and later vindicated are the
principle of equivalence and the principle of adiabatic invariance of the quantum number. Another of Einstein's
general principles, Mach's principle, is fiercely debated, and whether it holds in our world or not is still not
definitively established.

Theory of relativity and E = mc

Einstein's "Zur Elektrodynamik bewegter Korper" ("On the Electrodynamics of Moving Bodies") was received on
June 30, 1905 and published September 26 of that same year. It reconciles Maxwell's equations for electricity and
magnetism with the laws of mechanics, by introducing major changes to mechanics close to the speed of light. This
later became known as Einstein's special theory of relativity.

Consequences of this include the time-space frame of a moving body slowing down and contracting (in the direction
of motion) relative to the frame of the observer. This paper also argued that the idea of a luminiferous aether — one
of the leading theoretical entities in physics at the time — was superfluous.

In his paper on mass— energy equivalence, which had previously been considered to be distinct concepts, Einstein
deduced from his equations of special relativity what has been called the 20th century's best-known equation:
E= mc . This equation suggests that tiny amounts of mass could be converted into huge amounts of energy

and presaged the development of nuclear power. Einstein's 1905 work on relativity remained controversial for
many years, but was accepted by leading physicists, starting with Max Planck.

Photons

. [66] -

In a 1905 paper, Einstein postulated that light itself consists of localized particles (quanta). Einstein's light quanta
were nearly universally rejected by all physicists, including Max Planck and Niels Bohr. This idea only became
universally accepted in 1919, with Robert Millikan's detailed experiments on the photoelectric effect, and with the
measurement of Compton scattering.

Einstein concluded that each wave of frequency / is associated with a collection of photons with energy hf each,
where h is Planck's constant. He does not say much more, because he is not sure how the particles are related to the
wave. But he does suggest that this idea would explain certain experimental results, notably the photoelectric

effect

[67]

Albert Einstein 531

Quantized atomic vibrations

In 1907 Einstein proposed a model of matter where each atom in a lattice structure is an independent harmonic
oscillator. In the Einstein model, each atom oscillates independently - a series of equally spaced quantized states for
each oscillator. Einstein was aware that getting the frequency of the actual oscillations would be different, but he
nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could
solve the specific heat problem in classical mechanics. Peter Debye refined this model.

This work was the foundation of condensed matter physics.

Adiabatic principle and action-angle variables

Throughout the 1910s, quantum mechanics expanded in scope to cover many different systems. After Ernest
Rutherford discovered the nucleus and proposed that electrons orbit like planets, Niels Bohr was able to show that
the same quantum mechanical postulates introduced by Planck and developed by Einstein would explain the discrete
motion of electrons in atoms, and the periodic table of the elements.

Einstein contributed to these developments by linking them with the 1898 arguments Wilhelm Wien had made. Wien
had shown that the hypothesis of adiabatic invariance of a thermal equilibrium state allows all the blackbody curves
at different temperature to be derived from one another by a simple shifting process. Einstein noted in 1911 that the
same adiabatic principle shows that the quantity which is quantized in any mechanical motion must be an adiabatic
invariant. Arnold Sommerfeld identified this adiabatic invariant as the action variable of classical mechanics. The
law that the action variable is quantized was a basic principle of the quantum theory as it was known between 1900
and 1925.

Wave-particle duality

Although the patent office promoted Einstein to Technical Examiner Second Class in 1906, he had not given up on
academia. In 1908, he became a privatdozent at the University of Bern. In "iiber die Entwicklung unserer
Anschauungen iiber das Wesen und die Konstitution der Strahlung" ("The Development of Our Views on the
Composition and Essence of Radiation"), on the quantization of light, and in an earlier 1909 paper, Einstein showed
that Max Planck's energy quanta must have well-defined momenta and act in some respects as independent,
point-like particles. This paper introduced the photon concept (although the name photon was introduced later by
Gilbert N. Lewis in 1926) and inspired the notion of wave-particle duality in quantum mechanics.

Theory of critical opalescence

Einstein returned to the problem of thermodynamic fluctuations, giving a treatment of the density variations in a
fluid at its critical point. Ordinarily the density fluctuations are controlled by the second derivative of the free energy
with respect to the density. At the critical point, this derivative is zero, leading to large fluctuations. The effect of
density fluctuations is that light of all wavelengths is scattered, making the fluid look milky white. Einstein relates
this to Raleigh scattering, which is what happens when the fluctuation size is much smaller than the wavelength, and
which explains why the sky is blue. Einstein quantitatively derived critical opalescence from a treatment of
density fluctuations, and demonstrated how both the effect and Rayleigh scattering originate from the atomistic
constitution of matter.

Albert Einstein

532

Zero-point energy

Einstein's physical intuition led him to note that Planck's oscillator energies had
an incorrect zero point. He modified Planck's hypothesis by stating that the
lowest energy state of an oscillator is equal to / hf, to half the energy spacing
between levels. This argument, which was made in 1913 in collaboration with
Otto Stern, was based on the thermodynamics of a diatomic molecule which can
split apart into two free atoms.

General relativity and the Equivalence Principle

General relativity (GR) is a theory of gravitation that was developed by Albert
Einstein between 1907 and 1915. According to general relativity, the observed
gravitational attraction between masses results from the warping of space and
time by those masses. General relativity has developed into an essential tool in
modern astrophysics. It provides the foundation for the current understanding of
black holes, regions of space where gravitational attraction is so strong that not
even light can escape.

Einstein at the Solvay Conference in
1911

As Albert Einstein later said, the reason for the development of general relativity

was that the preference of inertial motions within special relativity was unsatisfactory, while a theory which from the

T711
outset prefers no state of motion (even accelerated ones) should appear more satisfactory. So in 1908 he

published an article on acceleration under special relativity. In that article, he argued that free fall is really inertial

motion, and that for a freefalling observer the rules of special relativity must apply. This argument is called the

Equivalence principle. In the same article, Einstein also predicted the phenomenon of gravitational time dilation. In

1911, Einstein published another article expanding on the 1907 article, in which additional effects such as the

deflection of light by massive bodies were predicted.

Hole argument and Entwurf theory

While developing general relativity, Einstein became confused about the gauge invariance in the theory. He
formulated an argument that led him to conclude that a general relativistic field theory is impossible. He gave up
looking for fully generally covariant tensor equations, and searched for equations that would be invariant under
general linear transformations only.

In June, 1913 the Entwurf ("draft") theory was the result of these investigations. As its name suggests, it was a
sketch of a theory, with the equations of motion supplemented by additional gauge fixing conditions. Simultaneously
less elegant and more difficult than general relativity, after more than two years of intensive work Einstein
abandoned the theory in November, 1915 after realizing that the hole argument was mistaken

[72]

Albert Einstein

533

Cosmology

In 1917, Einstein applied the General theory of relativity to model the structure of the universe as a whole. He

wanted the universe to be eternal and unchanging, but this type of universe is not consistent with relativity. To fix

this, Einstein modified the general theory by introducing a new notion, the cosmological constant. With a positive

[731
cosmological constant, the universe could be an eternal static sphere

Einstein believed a spherical static universe is philosophically preferred, because it would obey Mach's principle. He
had shown that general relativity incorporates Mach's principle to a certain extent in frame dragging by
gravitomagnetic fields, but he knew that Mach's idea would not work if space goes on forever. In a closed universe,
he believed that Mach's principle would hold.

Mach's principle has generated much controversy over the years.

Modern quantum theory

In 1917, at the height of his work on relativity, Einstein published an
article in Physikalische Zeitschrift that proposed the possibility of

stimulated emission, the physical process that makes possible the

T741
maser and the laser. This article showed that the statistics of

absorption and emission of light would only be consistent with

Planck's distribution law if the emission of light into a mode with n

photons would be enhanced statistically compared to the emission of

light into an empty mode. This paper was enormously influential in the

later development of quantum mechanics, because it was the first paper

to show that the statistics of atomic transitions had simple laws.

Einstein discovered Louis de Broglie's work, and supported his ideas,

which were received skeptically at first. In another major paper from

this era, Einstein gave a wave equation for de Broglie waves, which

Einstein suggested was the Hamilton— Jacobi equation of mechanics.

This paper would inspire Schrodinger's work of 1926.

Bose-Einstein statistics

In 1924, Einstein received a description of a statistical model from Indian physicist Satyendra Nath Bose, based on a
counting method that assumed that light could be understood as a gas of indistinguishable particles. Einstein noted
that Bose's statistics applied to some atoms as well as to the proposed light particles, and submitted his translation of
Bose's paper to the Zeitschrift fiir Physik. Einstein also published his own articles describing the model and its

implications, among them the Bose— Einstein condensate phenomenon that some particulates should appear at very

T751
low temperatures. It was not until 1995 that the first such condensate was produced experimentally by Eric Allin

Cornell and Carl Wieman using ultra-cooling equipment built at the NIST— JILA laboratory at the University of

Colorado at Boulder. Bose— Einstein statistics are now used to describe the behaviors of any assembly of bosons.

Einstein's sketches for this project may be seen in the Einstein Archive in the library of the Leiden University

[]

Albert Einstein 534

Energy momentum pseudotensor

General relativity includes a dynamical spacetime, so it is difficult to see how to identify the conserved energy and
momentum. Noether's theorem allows these quantities to be determined from a Lagrangian with translation
invariance, but general covariance makes translation invariance into something of a gauge symmetry. The energy
and momentum derived within general relativity by Noether's presecriptions do not make a real tensor for this
reason.

Einstein argued that this is true for fundamental reasons, because the gravitational field could be made to vanish by a
choice of coordinates. He maintained that the non-covariant energy momentum pseudotensor was in fact the best
description of the energy momentum distribution in a gravitational field. This approach has been echoed by Lev
Landau and Evgeny Lifshitz, and others, and has become standard.

The use of non-covariant objects like pseudotensors was heavily criticized in 1917 by Erwin Schrodinger and others.

Unified field theory

Following his research on general relativity, Einstein entered into a series of attempts to generalize his geometric

theory of gravitation to include electromagnetism as another aspect of a single entity. In 1950, he described his

[771
"unified field theory" in a Scientific American article entitled "On the Generalized Theory of Gravitation".

Although he continued to be lauded for his work, Einstein became increasingly isolated in his research, and his

efforts were ultimately unsuccessful. In his pursuit of a unification of the fundamental forces, Einstein ignored some

mainstream developments in physics, most notably the strong and weak nuclear forces, which were not well

understood until many years after his death. Mainstream physics, in turn, largely ignored Einstein's approaches to

unification. Einstein's dream of unifying other laws of physics with gravity motivates modern quests for a theory of

everything and in particular string theory, where geometrical fields emerge in a unified quantum-mechanical setting.

Wormholes

Einstein collaborated with others to produce a model of a wormhole. His motivation was to model elementary
particles with charge as a solution of gravitational field equations, in line with the program outlined in the paper "Do
Gravitational Fields play an Important Role in the Constitution of the Elementary Particles?". These solutions cut
and pasted Schwarzschild black holes to make a bridge between two patches.

If one end of a wormhole was positively charged, the other end would be negatively charged. These properties led
Einstein to believe that pairs of particles and antiparticles could be described in this way.

Einstein-Cartan theory

In order to incorporate spinning point particles into general relativity, the affine connection needed to be generalized
to include an antisymmetric part, called the torsion. This modification was made by Einstein and Cartan in the 1920s.

Equations of motion

The theory of general relativity has a fundamental law — the Einstein equations which describe how space curves,
the geodesic equation which describes how particles move may be derived from the Einstein equations.

Since the equations of general relativity are non-linear, a lump of energy made out of pure gravitational fields, like a
black hole, would move on a trajectory which is determined by the Einstein equations themselves, not by a new law.
So Einstein proposed that the path of a singular solution, like a black hole, would be determined to be a geodesic
from general relativity itself.

This was established by Einstein, Infeld, and Hoffmann for pointlike objects without angular momentum, and by
Roy Kerr for spinning objects.

Albert Einstein 535

Einstein's controversial beliefs in physics

In addition to his well-accepted results, some of Einstein's views are regarded as controversial:

• In the special relativity paper (in 1905), Einstein noted that, given a specific definition of the word "force" (a
definition which he later agreed was not advantageous), and if we choose to maintain (by convention) the
equation mass x acceleration = force, then one arrives at m/(l— -u 2 /c 2 )as the expression for the transverse mass of
a fast moving particle. This differs from the accepted expression today, because, as noted in the footnotes to
Einstein's paper added in the 1913 reprint, "it is more to the point to define force in such a way that the laws of
energy and momentum assume the simplest form", as was done, for example, by Max Planck in 1906, who gave
the now familiar expression ■m/\Jl—v 2 /c 2 iox the transverse mass. As Miller points out, this is equivalent to the
transverse mass predictions of both Einstein and Lorentz. Einstein had commented already in the 1905 paper that
"With a different definition of force and acceleration, we should naturally obtain other expressions for the masses.

T7S1

This shows that in comparing different theories... we must proceed very cautiously."

• Einstein published (in 1922) a qualitative theory of superconductivity based on the vague idea of electrons shared

in orbits. This paper predated modern quantum mechanics, and today is regarded as being incorrect. The current
theory of low temperature superconductivity was only worked out in 1957, thirty years after the establishing of
modern quantum mechanics. However, even today, superconductivity is not well understood, and alternative
theories continue to be put forward, especially to account for high-temperature superconductors.

• After introducing the concept of gravitational waves in 1917, Einstein subsequently entertained doubts about
whether they could be physically realized. In 1937 he published a paper saying that the focusing properties of
geodesies in general relativity would lead to an instability which causes plane gravitational waves to collapse in
on themselves. While this is true to a certain extent in some limits, because gravitational instabilities can lead to a
concentration of energy density into black holes, for plane waves of the type Einstein and Rosen considered in
their paper, the instabilities are under control. Einstein retracted this position a short time later.

• Einstein denied several times that black holes could form. In 1939 he published a paper that argues that a star
collapsing would spin faster and faster, spinning at the speed of light with infinite energy well before the point
where it is about to collapse into a black hole. This paper received no citations, and the conclusions are well
understood to be wrong. Einstein's argument itself is inconclusive, since he only shows that stable spinning
objects have to spin faster and faster to stay stable before the point where they collapse. But it is well understood
today (and was understood well by some even then) that collapse cannot happen through stationary states the way
Einstein imagined. Nevertheless, the extent to which the models of black holes in classical general relativity
correspond to physical reality remains unclear, and in particular the implications of the central singularity implicit
in these models are still not understood. Efforts to conclusively prove the existence of event horizons have still
not been successful.

• Closely related to his rejection of black holes, Einstein believed that the exclusion of singularities might restrict
the class of solutions of the field equations so as to force solutions compatible with quantum mechanics, but no
such theory has ever been found.

• In the early days of quantum mechanics, Einstein tried to show that the uncertainty principle was not valid, but by
1927 he had become convinced that it was valid.

• In the EPR paper, Einstein argued that quantum mechanics cannot be a complete realistic and local representation
of phenomena, given specific definitions of "realism", "locality", and "completeness". The modern consensus is
that Einstein's concept of realism is too restrictive.

• Einstein himself considered the introduction of the cosmological term in his 1917 paper founding cosmology as a
"blunder". The theory of general relativity predicted an expanding or contracting universe, but Einstein wanted
a universe which is an unchanging three dimensional sphere, like the surface of a three dimensional ball in four
dimensions. He wanted this for philosophical reasons, so as to incorporate Mach's principle in a reasonable way.
He stabilized his solution by introducing a cosmological constant, and when the universe was shown to be
expanding, he retracted the constant as a blunder. This is not really much of a blunder — the cosmological constant

Albert Einstein 536

is necessary within general relativity as it is currently understood, and it is widely believed to have a nonzero
value today.

• Einstein did not immediately appreciate the value of Minkowski's four-dimensional formulation of special
relativity, although within a few years he had adopted it as the basis for his theory of gravitation.

• Finding it too formal, Einstein believed that Heisenberg's matrix mechanics was incorrect. He changed his mind
when Schrodinger and others demonstrated that the formulation in terms of the Schrodinger equation, based on
Einstein's wave-particle duality was equivalent to Heisenberg's matrices.

Collaboration with other scientists

In addition to long time collaborators Leopold Infeld, Nathan Rosen, Peter Bergmann and others, Einstein also had
some one-shot collaborations with various scientists.

Einstein-de Haas experiment

Einstein and De Haas demonstrated that magnetization is due to the motion of electrons, nowadays known to be the
spin. In order to show this, they reversed the magnetization in an iron bar suspended on a torsion pendulum. They
confirmed that this leads the bar to rotate, because the electron's angular momentum changes as the magnetization
changes. This experiment needed to be sensitive, because the angular momentum associated with electrons is small,
but it definitively established that electron motion of some kind is responsible for magnetization.

Schrodinger gas model

Einstein suggested to Erwin Schrodinger that he might be able to reproduce the statistics of a Bose— Einstein gas by
considering a box. Then to each possible quantum motion of a particle in a box associate an independent harmonic
oscillator. Quantizing these oscillators, each level will have an integer occupation number, which will be the number
of particles in it.

This formulation is a form of second quantization, but it predates modern quantum mechanics. Erwin Schrodinger
applied this to derive the thermodynamic properties of a semiclassical ideal gas. Schrodinger urged Einstein to add
his name as co-author, although Einstein declined the invitation.

Einstein refrigerator

In 1926, Einstein and his former student Leo Szilard co-invented (and in 1930, patented) the Einstein refrigerator.

roil

This absorption refrigerator was then revolutionary for having no moving parts and using only heat as an input.
On 1 1 November 1930, U.S. Patent 1781541 was awarded to Albert Einstein and Leo Szilard for the refrigerator.
Their invention was not immediately put into commercial production, as the most promising of their patents were
quickly bought up by the Swedish company Electrolux to protect its refrigeration technology from competition.

Albert Einstein

537

Bohr versus Einstein

In the 1920s, quantum mechanics developed into a more complete theory.
Einstein was unhappy with the Copenhagen interpretation of quantum theory
developed by Niels Bohr and Werner Heisenberg, both in its outcomes and its
instrumentalist methodology, Einstein being a scientific realist. In this
interpretation, quantum phenomena are inherently probabilistic, with definite
states resulting only upon interaction with classical systems. A public debate
between Einstein and Bohr followed, lasting on and off for many years
(including during the Solvay Conferences). Einstein formulated thought
experiments against the Copenhagen interpretation, which were all rebutted by
Bohr. In a 1926 letter to Max Born, Einstein wrote: "I, at any rate, am convinced
that He [God] does not throw dice

.,[84]

Einstein and Niels Bohr, 1925

Einstein was never satisfied by what he perceived to be quantum theory's
intrinsically incomplete description of nature, and in 1935 he further explored the
issue in collaboration with Boris Podolsky and Nathan Rosen, noting that the
theory seems to require non-local interactions; this is known as the EPR paradox. The EPR experiment has since

been performed, with results confirming quantum theory's predictions
debate have found their way into philosophical discourse.

[861

Repercussions of the Einstein— Bohr

Einstein-Podolsky-Rosen paradox

In 1935, Einstein returned to the question of quantum mechanics. He considered how a measurement on one of two
entangled particles would affect the other. He noted, along with his collaborators, that by performing different
measurements on the distant particle, either of position or momentum, different properties of the entangled partner
could be discovered without disturbing it in any way.

He then used a hypothesis of local realism to conclude that the other particle had these properties already
determined. The principle he proposed is that if it is possible to determine what the answer to a position or
momentum measurement would be, without in any way disturbing the particle, then the particle actually has values
of position or momentum.

This principle distilled the essence of Einstein's objection to quantum mechanics. As a physical principle, it has since
been shown to be incompatible with experiments.

Albert Einstein

538

Political and religious views

Albert Einstein's political views emerged publicly in the middle of the
20th century due to his fame and reputation for genius. Einstein offered
to and was called on to give judgments and opinions on matters often
unrelated to theoretical physics or mathematics, (see main article)

Einstein's views on religious belief have been collected from
interviews and original writings. These views covered theological
determinism, agnosticism, humanism along with ethical culture, opting
for Spinoza's god over belief in a personal god, religious belief,
enlightenment and liberation, Jews, Christianity, Jesus, Pope Pius XII,
and the Catholic Church, (see main article)

Albert Einstein, seen here with his wife Elsa

Einstein and Zionist leaders, including future

President of Israel Chaim Weizmann, his wife Dr.

Vera Weizmann, Menahem Ussishkin, and

Ben-Zion Mossinson on arrival in New York City

in 1921.

Non-scientific legacy

While travelling, Einstein wrote daily to his wife Elsa and adopted
stepdaughters Margot and Use. The letters were included in the papers
bequeathed to The Hebrew University. Margot Einstein permitted the personal letters to be made available to the

T871

public, but requested that it not be done until twenty years after her death (she died in 1986 ). Barbara Wolff, of
The Hebrew University's Albert Einstein Archives, told the BBC that there are about 3,500 pages of private
correspondence written between 1912 and 1955

[88]

Einstein bequeathed the royalties from use of his image to The Hebrew University of Jerusalem. Corbis, successor to

The Roger Richman Agency, licenses the use of his name and associated imagery, as agent for the university

[89] [90]

In popular culture

In the period before World War II, Einstein was so well-known in America that he would be stopped on the street by

people wanting him to explain "that theory". He finally figured out a way to handle the incessant inquiries. He told

[91]
his inquirers "Pardon me, sorry! Always I am mistaken for Professor Einstein."

[92]
Einstein has been the subject of or inspiration for many novels, films, plays, and works of music. He is a favorite

model for depictions of mad scientists and absent-minded professors; his expressive face and distinctive hairstyle

have been widely copied and exaggerated. Time magazine's Frederic Golden wrote that Einstein was "a cartoonist's

dream come true'

[93]

Albert Einstein

539

Awards and honors

[94]

In 1922, Einstein was awarded the 1921 Nobel Prize in Physics,
"for his services to Theoretical Physics, and especially for his
discovery of the law of the photoelectric effect". This refers to his 1905
paper on the photoelectric effect, "On a Heuristic Viewpoint
Concerning the Production and Transformation of Light", which was
well supported by the experimental evidence by that time. The
presentation speech began by mentioning "his theory of relativity
[which had] been the subject of lively debate in philosophical circles
[and] also has astrophysical implications which are being rigorously
examined at the present time". (Einstein 1923)

It was long reported that, in accord with the divorce settlement, the
Nobel Prize money had been deposited in a Swiss bank account for
Marie to draw on the interest for herself and their two sons, while she
could only use the capital by agreement with Einstein. However,
personal correspondence made public in 2006 shows that he
invested much of it in the United States, and saw much of it wiped out
in the Great Depression. However, ultimately he paid Marie more

money than he received with the prize

[97]

In 1929, Max Planck presented Einstein with the Max Planck medal of
the German Physical Society in Berlin, for extraordinary achievements
in theoretical physics

[98]

In 1936, Einstein was awarded the Franklin Institute's Franklin Medal
for his extensive work on relativity and the photo-electric effect.

The International Union of Pure and Applied Physics named 2005 the

"World Year of Physics" in commemoration of the 100th anniversary

[99]
of the publication of the annus mirabilis papers.

The Albert Einstein Science Park is located on the hill Telegrafenberg
in Potsdam, Germany. The best known building in the park is the
Einstein Tower which has a bronze bust of Einstein at the entrance.
The Tower is an astrophysical observatory that was built to perform

checks of Einstein's theory of General Relativity

[100]

The Albert Einstein Memorial in central Washington, D.C. is a
monumental bronze statue depicting Einstein seated with manuscript
papers in hand. The statue, commissioned in 1979, is located in a grove
of trees at the southwest corner of the grounds of the National
Academy of Sciences on Constitution Avenue.

The chemical element 99, einsteinium, was named for him in August
1955, four months after Einstein's death. 2001 Einstein is an

inner main belt asteroid discovered on 5 March 1973

[103]

mmmmmrmmm

1879-1955

Israeli postage stamp (1956).

U.S. postage stamp (1966).

r\\\

f, , ■

AJ-lhBEPT

/f . GU+<4$jLu^>

[93] [104]

Soviet postage stamp (1979).

In 1999 Time magazine named him the Person of the Century,

ahead of Mahatma Gandhi and Franklin Roosevelt, among others. In the words of a biographer, "to the scientifically

[105]

Also in 1999, an opinion poll of 100

literate and the public at large, Einstein is synonymous with genius"

leading physicists ranked Einstein the "greatest physicist ever". A Gallup poll recorded him as the fourth most

Albert Einstein 540

admired person of the 20th century in the U.S.

In 1990, his name was added to the Walhalla temple for "laudable and distinguished Germans", which is located
east of Regensburg, in Bavaria, Germany.

The United States Postal Service honored Einstein with a Prominent Americans series (1965—1978) 80 postage
stamp.

Awards named after Einstein

The Albert Einstein Award (sometimes called the Albert Einstein Medal because it is accompanied with a gold
medal) is an award in theoretical physics, established to recognize high achievement in the natural sciences. It was
endowed by the Lewis and Rosa Strauss Memorial Fund in honor of Albert Einstein's 70th birthday. It was first
awarded in 1951 and included a prize money of $15,000, which was later reduced to $5,000. The

winner is selected by a committee (the first of which consisted of Einstein, Oppenheimer, von Neumann and
Weyl ) of the Institute for Advanced Study, which administers the award.

The Albert Einstein Medal is an award presented by the Albert Einstein Society in Bern, Switzerland. First given in
1979, the award is presented to people who have "rendered outstanding services" in connection with Einstein.

The Albert Einstein Peace Prize is given yearly by the Chicago, Illinois-based Albert Einstein Peace Prize
Foundation. Winners of the prize receive $50,000.

Publications

The following publications by Albert Einstein are referenced in this article. A more complete list of his
publications may be found at List of scientific publications by Albert Einstein.

• Einstein, Albert (1901), "Folgerungen aus den Capillaritatserscheinungen (Conclusions Drawn from the
Phenomena of Capillarity)", Annalen der Physik 4:513, doi:10.1002/andp.l9013090306

• Einstein, Albert (1905a), "Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen
Gesichtspunkt (On a Heuristic Viewpoint Concerning the Production and Transformation of Light)"

Annalen der Physik 17: 132—148, doi: 10. 1002/andp. 19053220607 This annus mirabilis paper on the photoelectric
effect was received by Annalen der Physik 18 March.

• Einstein, Albert (1905b), A new determination of molecular dimensions. This PhD thesis was completed 30 April
and submitted 20 July.

• Einstein, Albert (1905c), "On the Motion — Required by the Molecular Kinetic Theory of Heat — of Small
Particles Suspended in a Stationary Liquid", Annalen der Physik 17: 549—560. This annus mirabilis paper on
Brownian motion was received 1 1 May.

• Einstein, Albert (1905d), "On the Electrodynamics of Moving Bodies", Annalen der Physik 17: 891—921. This
annus mirabilis paper on special relativity was received 30 June.

• Einstein, Albert (1905e), "Does the Inertia of a Body Depend Upon Its Energy Content?", Annalen der Physik 18:
639—641. This annus mirabilis paper on mass-energy equivalence was received 27 September.

• Einstein, Albert (1915), "Die Feldgleichungen der Gravitation (The Field Equations of Gravitation)", Koniglich
Preussische Akademie der Wissenschaften: 844—847

• Einstein, Albert (1917a), "Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie (Cosmological
Considerations in the General Theory of Relativity)", Koniglich Preussische Akademie der Wissenschaften

• Einstein, Albert (1917b), "Zur Quantentheorie der Strahlung (On the Quantum Mechanics of Radiation)",
Physikalische Zeitschrift 18: 121—128

• Einstein, Albert (11 July 1923), "Fundamental Ideas and Problems of the Theory of Relativity" , Nobel
Lectures, Physics 1901—1921, Amsterdam: Elsevier Publishing Company, retrieved 25 March 2007

Albert Einstein 541

• Einstein, Albert (1924), "Quantentheorie des einatomigen idealen Gases (Quantum theory of monatomic ideal
gases)", Sitzungsberichte der Preussichen Akademie der Wissenschaften Physikalisch-Mathematische Klasse:
261—267. First of a series of papers on this topic.

• Einstein, Albert (1926), "Die Ursache der Maanderbildung der Flusslaufe und des sogenannten Baerschen
Gesetzes", Die Naturwissenschaften 14: 223—224, doi:10.1007/BF01510300. On Baer's law and meanders in the
courses of rivers.

• Einstein, Albert; Podolsky, Boris; Rosen, Nathan (15 May 1935), "Can Quantum-Mechanical Description of
Physical Reality Be Considered Complete?", Physical Review 47 (10): 777-780, doi:10.1 103/PhysRev.47.777

• Einstein, Albert (1940), "On Science and Religion", Nature (Edinburgh: Scottish Academic) 146: 605,
doi:10.1038/146605a0, ISBN 0707304539

• Einstein, Albert et al. (4 December 1948), "To the editors" , New York Times (Melville, NY: AIP, American
Inst, of Physics), ISBN 0735403597

• Einstein, Albert (May 1949), "Why Socialism?" [120] , Monthly Review, retrieved 16 January 2006

• Einstein, Albert (1950), "On the Generalized Theory of Gravitation", Scientific American CLXXXII (4): 13—17

• Einstein, Albert (1954), Ideas and Opinions, New York: Random House, ISBN 0-517-00393-7

• Einstein, Albert (1969) (in German), Albert Einstein, Hedwig und Max Born: Briefwechsel 1916—1955, Munich:
Nymphenburger Verlagshandlung, ISBN 388682005X

• Einstein, Albert (1979), Autobiographical Notes, Paul Arthur Schilpp (Centennial ed.), Chicago: Open Court,
ISBN 0-875-48352-6. The chasing a light beam thought experiment is described on pages 48—51.

• Collected Papers: Stachel, John, Martin J. Klein, a. J. Kox, Michel Janssen, R. Schulmann, Diana Komos
Buchwald and others (Eds.) (1987—2006), The Collected Papers of Albert Einstein, Vol. 1—10 , Princeton
University Press Further information about the volumes published so far can be found on the webpages of the

M221 r 1231

Einstein Papers Project and on the Princeton University Press Einstein Page

Notes

[I] Hans-Josef, Kupper (2000). "Various things about Albert Einstein" (http://www.einstein-website.de/z_information/variousthings.html).
einstein-website.de. . Retrieved 18 July 2009.

[2] Zahar, Elie (2001). Poincare's Philosophy. From Conventionalism to Phenomenology (http://books. google. com/?id=jJ12JAqvoSAC).

Carus Publishing Company, p. 41. ISBN 0-8126-9435-X. ., Chapter 2, p. 41 (http://books.google.com/books?id=jJ12JAqvoSAC&

pg=PA41).
[3] "The Nobel Prize in Physics 1921" (http://www.webcitation.org/5bLXMHV0). Nobel Foundation. Archived from the original (http://

nobelprize.org/nobel_prizes/physics/laureates/1921/) on 5 October 2008. . Retrieved 6 March 2007.
[4] "Einstein Biography" (http://nobelprize.org/nobel_prizes/physics/laureates/1921/einstein.html) Nobelprize.org.
[5] Paul Arthur Schilpp, editor (1951). Albert Einstein: Philosopher-Scientist, Volume II. New York: Harper and Brothers Publishers (Harper

Torchbook edition), pp. 730— 746. His non-scientific works include: About Zionism: Speeches and Lectures by Professor Albert Einstein

(1930), "Why War?" (1933, co-authored by Sigmund Freud), The World As I See It (1934), Out of My Later Years (1950), and a book on

science for the general reader, The Evolution of Physics (1938, co-authored by Leopold Infeld).
[6] WordNet for Einstein (http://wordnetweb.princeton.edu/perl/webwn?s=Einstein).
[7] "Albert Einstein — Biography" (http://nobelprize.org/nobel_prizes/physics/laureates/1921/einstein-bio.html). Nobel Foundation. .

Retrieved 7 March 2007.
[8] John J. Stachel (2002). Einstein from "B" to "Z" (http://books.google.com/books?id=OAsQ_hFjhrAC&pg=PA59). Springer, pp. 59-61.

ISBN 9780817641436. . Retrieved 20 February 2011.
[9] Einstein's Alleged Handicaps: The Legend of the Dull-Witted Child Who Grew Up to Be a Genius (http://www.albert-einstein.org/.

indexll.html)
[10] Schilpp (Ed.), P. A. (1979). Albert Einstein —Autobiographical Notes. Open Court Publishing Company, pp. 8—9.

[II] Dudley Herschbach, "Einstein as a Student", Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA,
page 3, web: HarvardChem-Einstein-PDF (http://www.chem.harvard.edu/herschbach/Einstein_Student.pdf): Max Talmud visited on
Thursdays for six years.

[12] "Albert's intellectual growth was strongly fostered at home. His mother, a talented pianist, ensured the children's musical education. His
father regularly read Schiller and Heine aloud to the family. Uncle Jakob challenged Albert with mathematical problems, which he solved with
'a deep feeling of happiness'. Most remarkable was Max Talmud, a poor Jewish medical student from Poland, 'for whom the Jewish
community had obtained free meals with the Einstein family'. Talmud came on Thursday nights for about six years, and 'invested his whole

Albert Einstein 542

person in examining everything that engaged [Albert's] interest'. Talmud had Albert read and discuss many books with him. These included a
series of twenty popular science books that convinced Albert 'a lot in the Bible stories could not be true', and a textbook of plane geometry that
launched Albert on avid self-study of mathematics, years ahead of the school curriculum. Talmud even had Albert read Kant; as a result
Einstein began preaching to his schoolmates about Kant, with 'forcefulness'." Einstein as a Student (http://www.chem.harvard.edu/
herschbach/Einstein_Student.pdf). (Herschbach, Dudley), from "Einstein for the 21st Century", Princeton University Press, 2008. Retrieved
18 April 2011.

[13] "Einstein's greatest intellectual stimulation came from a poor student who dined with his family once a week. It was an old Jewish custom to
take in a needy religious scholar to share the Sabbath meal; the Einsteins modified the tradition by hosting instead a medical student on
Thursdays. His name was Max Talmud, and he began his weekly visits when he was 21 and Einstein was 10." "Einstein & Faith" (http://
www.time.com/time/magazine/article/0,9171, 1607298-1, 00.html). Time.com. 2007-04-05. . Retrieved 2011-04-18.

[14] Mehra, Jagdish (2001). "Albert Einstein's first paper" (http://www.worldscibooks.com/phy_etextbook/4454/4454_chapl.pdf) (PDF).
The Golden Age of Physics. World Scientific. ISBN 9810249853. . Retrieved 4 March 2007.

[15] Highfield, Roger; Carter, Paul (1993). The Private Lives of Albert Einstein. London: Faber and Faber. p. 21. ISBN 0-571-17170-2.

[16] Highfield & Carter (1993, pp. 21,31,56-57)

[17] Albert Einstein Collected Papers, vol. 1, 1987, doc. 67.

[18] Troemel-Ploetz, D., "Mileva Einstein-Marie: The Woman Who Did Einstein's Mathematics", Women's Studies Int. Forum, vol. 13, no. 5, pp.
415-432, 1990.

[19] Walker, Evan Harris (February 1989) (PDF). Did Einstein Espouse his Spouse's Ideas? (http://philosci40.unibe.ch/lehre/winter99/
einstein/Walker_Stachel.pdf). Physics Today. . Retrieved 2011-03-31.

[20] Pais, A., Einstein Lived Here, Oxford University Press, 1994, pp. 1-29.

[21] Holton, G., Einstein, History, and Other Passions, Harvard University Press, 1996, pp. 177-193.

[22] Stachel, J., Einstein from B to Z, Birkhauser, 2002, pp. 26-38; 39-55. http://philoscience.unibe.ch/lehre/winter99/einstein/Stachell966.
pdf

[23] Martinez, A. A., "Handling evidence in history: the case of Einstein's Wife." School Science Review, 86 (3 16), March 2005, pp. 49-56. http:/
/www. ase.org.uk/htm/members_area/journals/ssr/ssr_march_05pdf/ eins_wife-pg49.pdf

[24] This conclusion is from Einstein's correspondence with Marie. Lieserl is first mentioned in a letter from Einstein to Marie (who was staying
with her family in or near Novi Sad at the time of Lieserl's birth) dated 4 February 1902 (Collected papers Vol. 1, document 134).

[25] Albrecht Folsing (1998). Albert Einstein: A Biography. Penguin Group. ISBN 0140237194; see section I, II,

[26] Highfield & Carter 1993, p. 216

[27] Now the Swiss Federal Institute of Intellectual Property (http://www.ipi. ch/E/institut/il.shtm), , retrieved 16 October 2006. See also
their FAQ about Einstein and the Institute (http://www.ipi. ch/E/institut/il094.shtm),

[28] Peter Galison, "Einstein's Clocks: The Question of Time" Critical Inquiry 26, no. 2 (Winter 2000): 355—389.

[29] Gallison, Question of Time.

[30] Galison, Peter (2003), Einstein's Clocks, Poincare's Maps: Empires of Time, New York: W.W. Norton, ISBN 0393020010

[31] (Einstein 1905b)

[32] "Universitat Zurich: Geschichte" (http://www.uzh.ch/about/portrait/history.html). Uzh.ch. 2010-12-02. . Retrieved 2011-04-03.

[33] Kant, Horst. "Albert Einstein and the Kaiser Wilhelm Institute for Physics in Berlin", in Renn, Jurgen. "Albert Einstein — Chief Engineer of
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[35] Heilbron, 2000, p. 84.

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[37] Hoffman and Dukas (1972), pp. 145-148; Folsing (1997), pp. 499-508.

[38] Isaacson, Walter. Einstein: His Life and Universe, Simon & Schuster (2007)

[39] Isaacson, Walter. Einstein: His Life and Universe, Simon & Schuster (2007) pp. 407-410

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[41] Hoffman, B. (1972), pp. 165-171; Folsing, A. (1997), pp. 666-677.

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[46] Stern, Fritz. Essay, "Einstein's Germany" (http://books. google. com/books?id=zGzcV40b3IkC&pg=PA97&dq=E=+Einstein's+

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[114] Sigmund, Dawson, Muhlberger (2006), Kurt Godel: The Album, Wiesbaden: Vieweg, ISBN 3834801739

[115] Albert Einstein Society in Bern (http://www.einstein-website.de/z_information/einsteinsociety.html) retrieved 17 July 2010

[116] Pugwash Online (http://www.pugwash.org/about/history.htm), , retrieved 20 December 2009

[117] http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf

[118] http://nobelprize.org/nobel_prizes/physics/laureates/1921/einstein-lecture.pdf

[119] http ://phy s4. harvard. edu/~ wilson/NYTimes 1 948.html

[120] http://www.monthlyreview.org/598einst.htm

[121] http://press.princeton.edU/einstein/writings.html#papers

[122] http://www.einstein.caltech.edu/index.html

[123] http://press.princeton.edu/einstein/

References
Further reading

• Folsing, Albrecht (1997): Albert Einstein: A Biography. New York: Penguin Viking. (Translated and abridged
from the German by Ewald Osers.)

• Hoffmann, Banesh, with the collaboration of Helen Dukas (1972): Albert Einstein: Creator and Rebel. London:
Hart-Davis, MacGibbon Ltd.

• Isaacson, Walter (2007): Einstein: His Life and Universe. Simon & Schuster Paperbacks, New York. ISBN
9780743264730

• Moring, Gary (2004): The complete idiot's guide to understanding Einstein (http://books.google.com/
books?id=875TTxildJ0C&dq=idiots+guide+to+einstein&printsec=frontcover&source=bl&
ots=W9rxRk0Ukn&sig=gbJach7BrzngSiFjODx95k8elDU&hl=en&sa=X&oi=book_result&resnum=6&
ct=result) ( 1st ed. 2000). Indianapolis IN: Alpha books (Macmillan USA). ISBN 0028631803

• Pais, Abraham (1982): Subtle is the Lord: The science and the life of Albert Einstein. Oxford University Press.
The definitive biography to date.

• Pais, Abraham (1994): Einstein Lived Here. Oxford University Press.

• Parker, Barry (2000): Einstein's Brainchild. Prometheus Books. A review of Einstein's career and
accomplishments, written for the lay public.

• Schweber, Sylvan S. (2008): Einstein and Oppenheimer: The Meaning of Genius. Harvard University Press.
ISBN 978-0674028289.

• Oppenheimer, J.R. (1971): "On Albert Einstein," p. 8-12 in Science and synthesis: an international colloquium
organized by Unesco on the tenth anniversary of the death of Albert Einstein and Teilhard de Chardin,
Springer- Verlag, 1971, 208 pp. (Lecture delivered at the UNESCO House in Paris on December 13, 1965.) Also
published in The New York Review of Books, March 17, 1966, On Albert Einstein by Robert Oppenheimer
(http://www.nybooks.com/articles/archives/1966/mar/17/on-albert-einstein/?pagination=false)

External links

• Works by Albert Einstein (public domain in Canada)

• The MacTutor History of Mathematics archive (http://www-history.mcs.st-andrews.ac.uk/Biographies/
Einstein.html), School of Mathematics and Statistics, University of St Andrews, Scotland, April 1997, retrieved
14 June 2009

• Why Socialism? (http://www.monthlyreview.org/598einstein.php) by Albert Einstein, Monthly Review, May
1949

• Nobelprize.org Biography: Albert Einstein (http://nobelprize.org/nobel_prizes/physics/laureates/1921/
einstein-bio. html)

Albert Einstein 546

• The Einstein You Never Knew (http://www.life.com/image/first/in-gallery/41492/
the-einstein-you-never-knew) - slideshow by Life magazine

• Albert Einstein (http://www.history.com/topics/albert-einstein)— Watch Videos

• Science Odyssey People And Discoveries (http://www.pbs.org/wgbh/aso/databank/entries/bpeins.html)

• MIT OpenCourseWare STS.042J / 8.225J : Einstein, Oppenheimer, Feynman: Physics in the 20th Century (http:/
/ocw. mit.edu/courses/science-technology-and-society/

sts-042j -einstein-oppenheimer-feynman-phy sics-in-the-20th-century-spring-2006/) Free, independent study
course that explores the changing roles of physics and physicists during the 20th century.

Emmy Noether

547

Emmy Noether

jOoU 15'

It p*

j i ^"

Amalie Emmy Noether

Born

23 March 1882Erlangen, Bavaria, Germany

Died

14 April 1935 (aged 53)Bryn Mawr, Pennsylvania

USA

Citizenship

Germany

Fields

Mathematics and Physics

Institutions

University of Gottingen
Bryn Mawr College

Alma mater

University of Erlangen

Doctoral advisor

Paul Gordan

Doctoral students

Max Deuring
Hans Fitting
Grete Hermann
Zeng Jiongzhi
Jacob Levitzki
Hans Reichenbach
Ernst Witt

Known for

Abstract algebra
Theoretical physics

Amalie Emmy Noether, German pronunciation: ['n0:te], (23 March 1882 — 14 April 1935) was an influential German
mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by
David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she

revolutionized the theories of rings, fields, and algebras. In physics, Noether's theorem explains the fundamental

T31
connection between symmetry and conservation laws.

She was born to a Jewish family in the Bavarian town of Erlangen; her father was the mathematician Max Noether.
Emmy originally planned to teach French and English after passing the required examinations, but instead studied
mathematics at the University of Erlangen, where her father lectured. After completing her dissertation in 1907
under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven
years. In 1915 she was invited by David Hilbert and Felix Klein to join the mathematics department at the University
of Gottingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and
she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain
the rank of privatdozent.

Noether remained a leading member of the Gottingen mathematics department until 1933; her students were
sometimes called the "Noether boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and

Emmy Noether

548

soon became the leading expositor of Noether's ideas: her work was the foundation for the second volume of his
influential 1931 textbook, Moderne Algebra. By the time of her plenary address at the 1932 International Congress
of Mathematicians in Zurich, her algebraic acumen was recognized around the world. The following year, Germany's
Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a
position at Bryn Mawr College in Pennsylvania. In 1935 she underwent surgery for an ovarian cyst and, despite
signs of a recovery, died four days later at the age of 53.

Noether's mathematical work has been divided into three "epochs". In the first (1908—1919), she made significant
contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the
calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever
proved in guiding the development of modern physics". In the second epoch, (1920—1926), she began work that
"changed the face of [abstract] algebra". In her classic paper Idealtheorie in Ringbereichen {Theory of Ideals in
Ring Domains, 1921) Noether developed the theory of ideals in commutative rings into a powerful tool with
wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are
named Noetherian in her honor. In the third epoch, (1927—1935), she published major works on noncommutative
algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and
ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of
research published by other mathematicians, even in fields far removed from her main work, such as algebraic
topology.

Biography

Emmy's father, Max Noether, was descended from a family of
wholesale traders in Germany. He had been paralyzed by poliomyelitis
at the age of fourteen. He regained mobility, but one leg remained
affected. Largely self-taught, he was awarded a doctorate from the
University of Heidelberg in 1868. After teaching there for seven years,
he took a position in the Bavarian city of Erlangen, where he met and
married Ida Amalia Kaufmann, the daughter of a prosperous
merchant. Max Noether's mathematical contributions were to
algebraic geometry mainly, following in the footsteps of Alfred
Clebsch. His best known results are the Brill— Noether theorem and the
residue, or AF+BG theorem; several other theorems are associated with him, including Max Noether's theorem

Noether grew up in the Bavarian city of Erlangen,
depicted here in a 1916 postcard

Emmy Noether was born on 23 March 1882, the first of four children. Her first name was "Amalie", after her mother
and paternal grandmother, but she began using her middle name at a young age. As a girl, she was well-liked. She
did not stand out academically although she was known for being clever and friendly. Emmy was near-sighted and
talked with a minor lisp during childhood. A family friend recounted a story years later about young Emmy quickly

ro]

solving a brain teaser at a children's party, showing logical acumen at that early age. Emmy was taught to cook and
clean — as were most girls of the time — and she took piano lessons. She pursued none of these activities with
passion, although she loved to dance.

Of her three brothers, only Fritz Noether, born in 1884, is remembered for his academic accomplishments. After
studying in Munich he made a reputation for himself in applied mathematics. Her eldest brother, Alfred, was born in
1883, was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. The youngest, Gustav
Robert, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928

[10]

Emmy Noether

549

University of Erlangen

Emmy Noether showed early proficiency in French and English. In the
spring of 1900 she took the examination for teachers of these
languages and received an overall score of sehr gut (very good). Her
performance qualified her to teach languages at schools reserved for
girls, but she chose instead to continue her studies at the University of
Erlangen.

This was an unconventional decision; two years earlier, the Academic
Senate of the university had declared that allowing coeducation would
"overthrow all academic order". One of only two women students in
a university of 986, Noether was forced to audit classes and required
the permission of individual professors whose lectures she wished to
attend. Despite the obstacles, on 14 July 1903 she passed the

ri2i

graduation exam at a Realgymnasium in Nuremberg.

Paul Gordan supervised Noether's doctoral
dissertation on invariants of biquadratic forms

During the 1903—04 winter semester, she studied at the University of Gottingen, attending lectures given by
astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David
Hilbert. Soon thereafter, restrictions on women's rights in that university were rescinded.

Noether returned to Erlangen. She officially reentered the university on 24 October 1904, and declared her intention
to focus solely on mathematics. Under the supervision of Paul Gordan she wrote her dissertation, Uber die Bildung

des Formensystems der terndren biquadratischen Form {On Complete Systems of Invariants for Ternary Biquadratic

1131
Forms, 1907). Although it had been well received, Noether later described her thesis as "crap".

For the next seven years (1908—1915) she taught at the University of Erlangen's Mathematical Institute without pay,
occasionally substituting for her father when he was too ill to lecture. In 1910 and 1911 she published an extension
of her thesis work from three variables to n variables.

Gordan retired in the spring of 1910, but continued to teach
occasionally with his successor, Erhard Schmidt, who left shortly
afterward for a position in Breslau. Gordan retired from teaching
altogether in 1911 with the arrival of his second successor, Ernst
Fischer. Gordan died in December 1912.

According to Hermann Weyl, Fischer was an important influence on
Noether, in particular by introducing her to the work of David
Hilbert. From 1913 to 1916 Noether published several papers
extending and applying Hilbert's methods to mathematical objects
such as fields of rational functions and the invariants of finite groups.
This phase marks the beginning of her engagement with abstract
algebra, the field of mathematics to which she would make
groundbreaking contributions.

Noether and Fischer shared lively enjoyment of mathematics and
would often discuss lectures long after they were over; Noether is
known to have sent postcards to Fischer continuing her train of
mathematical thoughts

[14]

1~T^ -TKr^ ~i/^, # %^ ^,-4 -~^,._ ryu^.

Noether sometimes used postcards to discuss

abstract algebra with her colleague, Ernst Fischer;

this card is postmarked 10 April 1915

University of Gottingen

Emmy Noether

550

In the spring of 1915, Noether was invited to return to the University of Gottingen by David Hilbert and Felix Klein.
Their effort to recruit her, however, was blocked by the philologists and historians among the philosophical faculty:
women, they insisted, should not become privatdozent. One faculty member protested: "What will our soldiers think
when they return to the university and find that they are required to learn at the feet of a woman?" Hilbert
responded with indignation, stating, "I do not see that the sex of the candidate is an argument against her admission
as privatdozent. After all, we are a university, not a bath house."

Noether left for Gottingen in late April; two weeks later her mother died
suddenly in Eriangen. She had previously received medical care for an eye
condition, but its nature and impact on her death is unknown. At about the same
time Noether's father retired and her brother joined the German Army to serve in
World War I. She returned to Eriangen for several weeks, mostly to care for her
aging father.

During her first years teaching at Gottingen she did not have an official position
and was not paid; her family paid for her room and board and supported her
academic work. Her lectures often were advertised under Hilbert's name, and
Noether would provide "assistance".

Soon after arriving at Gottingen, however, she demonstrated her capabilities by
proving the theorem now known as Noether's theorem, which shows that a

conservation law is associated with any differentiable symmetry of a physical

ri7i
system. American physicists Leon M. Lederman and Christopher T. Hill

argue in their book Symmetry and the Beautiful Universe that Noether's theorem

is "certainly one of the most important mathematical theorems ever proved in

guiding the development of modern physics, possibly on a par with the

Pythagorean theorem".

When World War I ended, the German Revolution of 1918—19
brought a significant change in social attitudes, including more
rights for women. In 1919 the University of Gottingen allowed
Noether to proceed with her habilitation (eligibility for tenure).
Her oral examination was held in late May, and she successfully
delivered her habilitation lecture in June.

Three years later she received a letter from the Prussian Minister
for Science, Art, and Public Education, in which he conferred on
her the title of nicht beamteter ausserordentlicher Professor (an

untenured professor with limited internal administrative rights

n si -
and functions ). This was an unpaid "extraordinary" professorship, not the higher "ordinary" professorship, which

was a civil-service position. Although it recognized the importance of her work, the position still provided no salary.

Noether was not paid for her lectures until she was appointed to the special position of Lehrauftrage fur Algebra a

i ,- [19]
year later.

In 1915 David Hilbert invited Emmy

Noether to join the mathematics

department at the University of

Gottingen, challenging the views of

some of his colleagues that a woman

should not be allowed to teach at a

university

The mathematics department at the University of

Gottingen allowed Noether's habilitation in 1919, four

years after she had begun lecturing at the school

Emmy Noether 55 1

Seminal work in abstract algebra

Although Noether's theorem had a profound effect upon physics, among mathematicians she is best remembered for
her seminal contributions to abstract algebra. As Nathan Jacobson says in his Introduction to Noether's Collected
Papers,

The development of abstract algebra, which is one of the most distinctive innovations of twentieth
century mathematics, is largely due to her — in published papers, in lectures, and in personal influence
on her contemporaries.

Noether's groundbreaking work in algebra began in 1920. In collaboration with W. Schmeidler, she then published a
paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she
published a landmark paper called, Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard

to ideals. A noted algebraist, Irving Kaplansky, has called this work "revolutionary", and the publication gave rise

T211
to the term "Noetherian ring" and several other mathematical objects being dubbed, Noetherian.

In 1924, a young Dutch mathematician, B. L. van der Waerden, arrived at the University of Gottingen. He
immediately began working with Noether, who provided invaluable methods of abstract conceptualization, van der

["221

Waerden later said that her originality was "absolute beyond comparison". In 1931 he published Moderne
Algebra, a central text in the field; its second volume borrowed heavily from Noether's work. Although Emmy

Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin

T231
and E. Noether". She sometimes allowed her colleagues and students to receive credit for her ideas, helping them

develop their careers at the expense of her own.

van der Waerden's visit was part of a convergence of mathematicians from all over the world to Gottingen, which
became a major hub of mathematical and physical research. From 1926 to 1930 the Russian topologist, Pavel
Alexandrov, lectured at the university, and he and Noether quickly became good friends. He began referring to her as
der Noether, using the masculine German article as a term of endearment to show his respect. She tried to arrange for
him to obtain a position at Gottingen as a regular professor, but was only able to help him secure a scholarship from
the Rockefeller Foundation. They met regularly and enjoyed discussions about the intersections of algebra and
topology. In his 1935 memorial address, Alexandrov named Emmy Noether "the greatest woman mathematician of
all time". [26]

Lecturing and students

In Gottingen, Noether supervised more than a dozen doctoral students; her first was Grete Hermann, who defended

T271
her dissertation in February 1925. She later spoke reverently of her "dissertation-mother". Noether also

supervised Max Deuring, who distinguished himself as an undergraduate and went on to contribute significantly to

the field of arithmetic geometry; Hans Fitting, remembered for Fitting's theorem and the Fitting lemma; and Zeng

Jiongzhi, who proved Tsen's theorem. She also worked closely with Wolfgang Krull, who greatly advanced

T2R1

commutative algebra with his Hauptidealsatz and his dimension theory for commutative rings.

In addition to her mathematical insight, Noether was respected for her consideration of others. Although she

sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant

helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to

[291
name her "a severe critic", but she combined this demand for accuracy with a nurturing attitude. A colleague later

described her this way: "Completely unegotistical and free of vanity, she never claimed anything for herself, but

promoted the works of her students above all."

Her frugal lifestyle at first was due to being denied pay for her work; however, even after the university began

paying her a small salary in 1923, she continued to live a simple and modest life. She was paid more generously later

nil
in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether.

Emmy Noether 552

Mostly unconcerned about appearance and manners, she focused on her studies to the exclusion of romance and
fashion. A distinguished algebraist Olga Taussky-Todd described a luncheon, during which Noether, wholly

engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped

[32]
it off from her dress, completely unperturbed". Appearance-conscious students cringed as she retrieved the

handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students

once approached her during a break in a two-hour class to express their concern, but were unable to break through

[331
the energetic mathematics discussion she was having with other students.

According to van der Waerden's obituary of Emmy Noether, she did not follow a lesson plan for her lectures, which
frustrated some students. Instead, she used her lectures as a spontaneous discussion time with her students, to think
through and clarify important cutting-edge problems in mathematics. Some of her most important results were
developed in these lectures, and the lecture notes of her students formed the basis for several important textbooks,
such as those of van der Waerden and Deuring.

Several of her colleagues attended her lectures, and she allowed some of her ideas, such as the crossed product

(verschranktes Produkt in German) of associative algebras, to be published by others. Noether was recorded as

[341
having given at least five semester-long courses at Gottingen:

• Winter 1924/25: Gruppentheorie und hyperkomplexe Zahlen (Group Theory and Hypercomplex Numbers)

• Winter 1927/28: Hyperkomplexe Grossen und Darstellungstheorie (Hypercomplex Quantities and Representation
Theory)

• Summer 1928: Nichtkommutative Algebra (Noncommutative Algebra)

• Summer 1929: Nichtkommutative Arithmetik (Noncommutative Arithmetic)

• Winter 1929/30: Algebra der hyperkomplexen Grossen (Algebra of Hypercomplex Quantities)

These courses often preceded major publications in these areas.

Noether spoke quickly — reflecting the speed of her thoughts, many said — and demanded great concentration from

her students. Students who disliked her style often felt alienated; one wrote in a notebook with regard to a class that

T351
ended at 1:00 pm: "It's 12:50, thank God!" Some pupils felt that she relied too much on spontaneous discussions.

Her most dedicated students, however, relished the enthusiasm with which she approached mathematics, especially

since her lectures often built on earlier work they had done together.

She developed a close circle of colleagues and students who thought along similar lines and tended to exclude those
who did not. "Outsiders" who occasionally visited Noether's lectures usually spent only 30 minutes in the room
before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been
defeated; he has cleared out."

Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the

building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and

T371
lectured at a local coffee house. Later, after she had been dismissed by the Third Reich, she invited students into

[30]
her home to discuss their future plans and mathematical concepts.

Emmy Noether

553

Moscow

In the winter of 1928—29 Noether accepted an invitation to Moscow
State University, where she continued working with P. S. Alexandrov.
In addition to carrying on with her research, she taught classes in
abstract algebra and algebraic geometry. She worked with the
topologists, Lev Pontryagin and Nikolai Chebotaryov, who later
praised her contributions to the development of Galois theory.

Although politics was not central to her life, Noether took a keen

interest in political matters and, according to Alexandrov, showed

considerable support for the Russian Revolution (1917). She was

especially happy to see Soviet advancements in the fields of science

and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This

attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student

leaders complained of living with "a Marxist-leaning Jewess"

Noether taught at the Moscow State University
during the winter of 1928—29

[40]

Noether planned to return to Moscow, an effort for which she received support
from Alexandrov. After she left Germany in 1933 he tried to help her gain a
chair at Moscow State University through the Soviet Education Ministry.
Although this effort proved unsuccessful, they corresponded frequently during
the 1930s, and in 1935 she made plans for a return to the Soviet Union.
Meanwhile her brother, Fritz accepted a position at the Research Institute for

Mathematics and Mechanics in Tomsk, in the Siberian Federal District of

T411
Russia, after losing his job in Germany.

Recognition

In 1932 Emmy Noether and Emil Artin received the Ackermann— Teubner

T421
Memorial Award for their contributions to mathematics. The prize carried a

monetary reward of 500 Reichsmarks and was seen as a long-overdue official

recognition of her considerable work in the field. Nevertheless, her colleagues

expressed frustration at the fact that she was not elected to the Gottingen Gesellschaft der Wissenschaften (academy

of sciences) and was never promoted to the position of Ordentlicher Professor (full professor).

Noether's colleagues celebrated her fiftieth birthday in 1932, in typical
mathematicians' style. Helmut Hasse dedicated an article to her in the
Mathematische Annalen, wherein he confirmed her suspicion that some
aspects of noncommutative algebra are simpler than those of
commutative algebra, by proving a noncommutative reciprocity

law

[44]

This pleased her immensely. He also sent her a mathematical

riddle, the "m|iv-riddle of syllables", which she solved immediately;

[431
the riddle has been lost.

Noether visited Zurich in 1932 to deliver a

plenary address at the International Congress of

Mathematicians

In November of the same year, Noether delivered a plenary address

{grofier Vortrag) on "Hyper-complex systems in their relations to

commutative algebra and to number theory" at the International

Congress of Mathematicians in Zurich. The congress was attended by 800 people, including Noether's colleagues

Hermann Weyl, Edmund Landau, and Wolfgang Krull. There were 420 official participants and twenty-one plenary

Emmy Noether 554

addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her

[451
contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.

Expulsion from Gottingen

When Adolf Hitler became the German Reichskanzler in January 1933, Nazi activity around the country increased
dramatically. At the University of Gottingen the German Student Association led the attack on the "un-German
spirit" and was aided by a privatdozent named Werner Weber, a former student of Emmy Noether. Antisemitic
attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students
want Aryan mathematics and not Jewish mathematics."

One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service

which removed Jews and politically suspect government employees (including university professors) from their jobs

unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933 Noether received

a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3

of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of

[471
Gottingen." Several of Noether's colleagues, including Max Born and Richard Courant, had their positions

T471
revoked. Noether accepted the decision calmly, providing support for others during this difficult time. Hermann

Weyl later wrote that "Emmy Noether — her courage, her frankness, her unconcern about her own fate, her

conciliatory spirit — was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral

solace." Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss

class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization

1471
Sturtnabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later.

Bryn Mawr

As dozens of newly unemployed professors began searching for

positions outside of Germany, their colleagues in the United States

sought to provide assistance and job opportunities for them. Albert

Einstein and Hermann Weyl were appointed by the Institute for

Advanced Study in Princeton, while others worked to find a sponsor

required for legal immigration. Noether was contacted by

representatives of two educational institutions, Bryn Mawr College in

the United States and Somerville College at the University of Oxford

in England. After a series of negotiations with the Rockefeller Bryn Mawr College provided a welcoming home

Foundation, a grant to Bryn Mawr was approved for Noether and she

for Noether during the last two years of her life

took a position there, starting in late 1933.

At Bryn Mawr, Noether met and befriended Anna Wheeler, who had studied at Gottingen just before Noether arrived

there. Another source of support at the college was the Bryn Mawr president, Marion Edwards Park, who

[49]
enthusiastically invited mathematicians in the area to "see Dr. Noether in action!" Noether and a small team of

students worked quickly through van der Waerden's 1930 book Moderne Algebra I and parts of Erich Hecke's

Theorie der algebraischen Zahlen {Theory of algebraic numbers, 1908).

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham
Flexner and Oswald Veblen. She also worked with and supervised Abraham Albert and Harry Vandiver.
However, she remarked about Princeton University that she was not welcome at the "men's university, where nothing

[52]

female is admitted".

Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her
favorite subjects. In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother Fritz

Emmy Noether

555

before he left for Tomsk. Although many of her former colleagues had been forced out of the universities, she was

[541
able to use the library as a "foreign scholar".

Death

In April 1935 doctors discovered a tumor in Noether's
pelvis. Worried about complications from surgery, they
ordered two days of bed rest first. During the operation
they discovered an ovarian cyst "the size of a large
cantaloupe". Two smaller tumors in her uterus
appeared to be benign and were not removed, to avoid
prolonging surgery. For three days she appeared to
convalesce normally, and recovered quickly from a
circulatory collapse on the fourth. On 14 April, she fell
unconscious, her temperature soared to 109 °F
(42.8 °C), and she died. "[I]t is not easy to say what had
occurred in Dr. Noether", one of the physicians wrote.
"It is possible that there was some form of unusual and
virulent infection, which struck the base of the brain
where the heat centers are supposed to be located."

Noether's remains were placed under the walkway surrounding the
cloisters of Bryn Mawr's M. Carey Thomas Library

A few days after Noether's death her friends and associates at Bryn Mawr held a small memorial service at President
Park's house. Hermann Weyl and Richard Brauer traveled from Princeton and spoke with Wheeler and Taussky
about their departed colleague. In the months which followed, written tributes began to appear around the globe:
Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov in paying their respects. Her body was
cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn

Mawr

,. [56]

Contributions to mathematics and physics

First and foremost Noether is remembered by mathematicians as an algebraist and for her work in topology.
Physicists appreciate her best for her famous theorem because of its far-ranging consequences for the study of

subatomic particles and dynamic systems. She showed an acute propensity for abstract thought, which allowed her to

1571
approach problems of mathematics in fresh and original ways. Her friend and colleague Hermann Weyl described

her scholarly output in three epochs:

"Emmy Noether's scientific production fell into three clearly distinct epochs:

(1) the period of relative dependence, 1907—1919;

(2) the investigations grouped around the general theory of ideals 1920—1926;

(3) the study of the non-commutative algebras, their representations by linear transformations, and their
application to the study of commutative number fields and their arithmetics." (Weyl 1935)

In the first epoch (1907—19), Noether dealt primarily with differential and algebraic invariants, beginning with her
dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and
abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to
Gordan, Ernst Sigismund Fischer. After moving to Gottingen in 1915, she produced her seminal work for physics,
the two Noether's theorems.

In the second epoch (1920—26), Noether devoted herself to developing the theory of mathematical rings

[58]

Emmy Noether 556

In the third epoch (1927—35), Noether focused on noncommutative algebra, linear transformations, and commutative
number fields.

Historical context

In the century from 1832 to Noether's death in 1935, the field of mathematics — specifically algebra — underwent a
profound revolution, whose reverberations are still being felt. Mathematicians of previous centuries had worked on
practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equations, as well as on the
related problem of constructing regular polygons using compass and straightedge. Beginning with Carl Friedrich
Gauss' 1829 proof that prime numbers such as five can be factored in Gaussian integers, Evariste Galois's
introduction of permutation groups in 1832 (although, because of his death, his papers were only published in 1846
by Liouville), William Rowan Hamilton's discovery of quaternions in 1843, and Arthur Cayley's more modern
definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by
ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this
new field, abstract algebra.

Abstract algebra and begriffliche Mathematik (conceptual mathematics)

Two of the most basic objects in abstract algebra are groups and rings.

A group consists of a set of elements and a single operation which combines a first and a second element and returns
a third. The operation must satisfy certain constraints for it to determine a group: It must be closed (when applied to
any pair of elements of the associated set, the generated element must also be a member of that set), it must be
associative, there must be an identity element (an element which, when combined with another element using the
operation, results in the original element, such as adding zero to a number or multiplying it by one), and for every
element there must be an inverse element.

A ring likewise, has a set of elements, but now has two operations. The first operation must make the set a group,
and the second operation is associative and distributive with respect to the first operation. It may or may not be
commutative; this means that the result of applying the operation to a first and a second element is the same as to the
second and first — the order of the elements does not matter. If every non-zero element has a multiplicative inverse
(an element x such that ax = xa = 1), the ring is called a division ring. A field is defined as a commutative division
ring.

Groups are frequently studied through group representations. In their most general form, these consist of a choice of
group, a set, and an action of the group on the set, that is, an operation which takes an element of the group and an
element of the set and returns an element of the set. Most often, the set is a vector space, and the group represents
symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. This is a
type of symmetry of space, because space itself does not change when it is rotated even though the positions of
objects in it do. Noether used these sorts of symmetries in her work on invariants in physics.

A powerful way of studying rings is through their modules. A module consists of a choice of ring, another set,
usually distinct from the underlying set of the ring and called the underlying set of the module, an operation on pairs
of elements of the underlying set of the module, and an operation which takes an element of the ring and an element
of the module and returns an element of the module. The underlying set of the module and its operation must form a
group. A module is a ring-theoretic version of a group representation: Ignoring the second ring operation and the
operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds
of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring
itself. An important special case of this is an algebra. (The word algebra means both a subject within mathematics as
well as an object studied in the subject of algebra.) An algebra consists of a choice of two rings and an operation
which takes an element from each ring and returns an element of the second ring. This operation makes the second
ring into a module over the first. Often the first ring is a field.

Emmy Noether 557

Words such as "element" and "combining operation" are very general, and can be applied to many real-world and
abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group
(or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and
multiplication, are just one example. For example, the elements might be computer data words, where the first
combining operation is exclusive or and the second is logical conjunction. Theorems of abstract algebra are powerful
because they are general; they govern many systems. It might be imagined that little could be concluded about
objects defined with so few properties, but precisely therein lay Noether's gift: to discover the maximum that could
be concluded from a given set of properties, or conversely, to identify the minimum set, the essential
properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions
by generalizing from known examples; rather, she worked directly with the abstractions. As van der Waerden
recalled in his obituary of her,

The maxim by which Emmy Noether was guided throughout her work might be formulated as follows:
"Any relationships between numbers, functions, and operations become transparent, generally
applicable, and fully productive only after they have been isolated from their particular objects and been
formulated as universally valid concepts.

This is the begriffliche Mathematik (purely conceptual mathematics) that was characteristic of Noether. This style of
mathematics was adopted by other mathematicians and, after her death, flowered into new forms, such as category
theory.

Integers as an example of a ring

The integers form a commutative ring whose elements are the integers, and the combining operations are addition
and multiplication. Any pair of integers can be added or multiplied, always resulting in another integer, and the first
operation, addition, is commutative, i.e., for any elements a and b in the ring, a + b = b + a. The second operation,
multiplication, also is commutative, but that need not be true for other rings, meaning that a combined with b might
be different from b combined with a. Examples of noncommutative rings include matrices and quaternions. The
integers do not form a division ring, because the second operation cannot always be inverted; there is no integer a
such that 3 x a - 1 .

The integers have additional properties which do not generalize to all commutative rings. An important example is
the fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime
numbers. Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem,
now called the Lasker— Noether theorem, for the ideals of many rings. Much of Noether's work lay in determining
what properties do hold for all rings, in devising novel analogs of the old integer theorems, and in determining the
minimal set of assumptions required to yield certain properties of rings.

Emmy Noether

558

First epoch (1908-19)

Algebraic invariant theory

Much of Noether's work in the first epoch of her career
was associated with invariant theory, principally
algebraic invariant theory. Invariant theory is
concerned with expressions that remain constant
(invariant) under a group of transformations. As an
everyday example, if a rigid yardstick is rotated, the
coordinates (x, y, z) of its endpoints change, but its

2 2 2 2

length L given by the formula L = Ax + A v + Az
remains the same. Invariant theory was an active area
of research in the later nineteenth century, prompted in
part by Felix Klein's Erlangen program, according to
which different types of geometry should be
characterized by their invariants under transformations,
e.g., the cross-ratio of projective geometry. The
archetypal example of an invariant is the discriminant

2 2 2

B - 4AC of a binary quadratic form Ax + Bxy + Cy .
This is called an invariant because it is unchanged by
linear substitutions x— >ax + by, y— >cx + dy with determinant ad - be = 1. These substitutions form the special linear
group SL . (There are no invariants under the general linear group of all invertible linear transformations because
these transformations can be multiplication by a scaling factor. To remedy this, classical invariant theory also
considered relative invariants, which were forms invariant up to a scale factor.) One can ask for all polynomials in
A, B, and C that are unchanged by the action of SL ; these are called the invariants of binary quadratic forms, and
turn out to be the polynomials in the discriminant. More generally, one can ask for the invariants of homogeneous

r Or

polynomials Ax y + ... + A x y of higher degree, which will be certain polynomials in the coefficients A , ... , A ,
and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.

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Table 2 from Noether's dissertation on invariant theory. This
table collects 202 of the 331 invariants of ternary biquadratic forms.

These forms are graded in two variables x and u. The horizontal
direction of the table lists the invariants with increasing grades in x,

while the vertical direction lists them with increasing grades in u.

One of the main goals of invariant theory was to solve the "finite basis problem". The sum or product of any two
invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting
with a finite list of invariants, called generators, and then, adding or multiplying the generators together. For
example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.
Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to
mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two
variables. He proved this by giving a constructive method for finding all of the invariants and their

generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890,
David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of
variables. Furthermore, his method worked, not only for the special linear group, but also for some of its

subgroups such as the special orthogonal group. His first proof caused some controversy because it did not give a
method for constructing the generators, although in later work he made his method constructive. For her thesis,
Noether extended Gordan's computational proof to homogeneous polynomials in three variables. Noether's
constructive approach made it possible to study the relationships among the invariants. Later, after she had turned to
more abstract methods, Noether called her thesis Mist (crap) and Formelngestriipp (a jungle of equations).

Emmy Noether 559

Galois theory

Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a
polynomial equation of a variable x of degree n, in which the coefficients are drawn from some ground field, which
might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not
be choices of x, which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. If the

polynomial is x + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of x
makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain
roots, and if it is extended enough, then it always has a number of roots equal to its degree. Continuing the previous
example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, i and -i, where i is the

imaginary unit, that is, i = -1. More generally, the extension field in which a polynomial can be factored into its
roots is known as the splitting field of the polynomial.

The Galois group of a polynomial is the set of all ways of transforming the splitting field, while preserving the
ground field and the roots of the polynomial. (In mathematical jargon, these transformations are called

automorphisms.) The Galois group of x + 1 consists of two elements: The identity transformation, which sends
every complex number to itself, and complex conjugation, which sends i to -i. Since the Galois group does not
change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots
unchanged. Each root can move to another root, however, so transformation determines a permutation of the n roots
among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory,
which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence
with the subgroups of the Galois group.

In 1918, Noether published a seminal paper on the inverse Galois problem. Instead of determining the Galois
group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it
always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to
"Noether's problem", which asks whether the fixed field of a subgroup G of the permutation group S acting on the
field k(x , ... , x ) always is a pure transcendental extension of the field k. (She first mentioned this problem in a 1913
paper, where she attributed the problem to her colleague Fischer.) She showed this was true for n = 2,3, or 4. In
1969, R. G. Swan found a counter-example to Noether's problem, with n = 47 and G a cyclic group of order 47
(although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem
remains unsolved.

Physics

Noether was brought to Gottingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant
theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by
Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, due
to the fact that gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a

fundamental tool of modern theoretical physics, with her first Noether's theorem, which she proved in 1915, but

T721
did not publish until 1918. She solved the problem not only for general relativity, but determined the conserved

quantities for every system of physical laws that possesses some continuous symmetry.

Upon receiving her work, Einstein wrote to Hilbert: "Yesterday I received from Miss Noether a very interesting

paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at

[731
Gottingen should take some lessons from Miss Noether! She seems to know her stuff."

For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws

that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of

T741
the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in

space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the

system is responsible for the conservation law. As another example, if a physical experiment has the same outcome

Emmy Noether 560

at any place and at any time, then its laws are symmetric under continuous translations in space and time; by
Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this
system, respectively.

Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives
into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the
conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of
a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical
phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon: if the theory
has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the
theory to be correct, this conservation must be observable in experiments.

Second epoch (1920-26)

Although the results of Noether's first epoch were impressive and useful, her fame as a mathematician rests more on
the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der
Waerden in their obituaries of her.

In these epochs, she was not merely applying ideas and methods of earlier mathematicians; rather, she was crafting
new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a
completely new theory of ideals in rings, generalizing earlier work of Richard Dedekind. She also is renowned for
developing ascending chain conditions, a simple finiteness condition that yielded powerful results in her hands. Such
conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a
new perspective, such as elimination theory and the algebraic varieties that had been studied by her father.

Ascending and descending chain conditions

In this epoch, Noether became famous for her deft use of ascending {Teilerkettensatz) or descending
{Vielfachenkettensatz) chain conditions. A sequence of non-empty subsets A , A , A , etc. of a set S is usually said to
be strictly ascending, if each is a subset of the next

Ai c A 2 C A 3 c ■ ■ ■

The ascending chain condition requires that such sequences break off after a finite number of steps; in other words,
all such sequences of subsets must be finite. Conversely, with strictly descending sequences of subsets

Ai D A 2 D A 3 D ■ ■ ■
the descending chain condition requires that such sequences break off after a finite number.

Ascending and descending chain conditions are general, meaning that they can be applied to many types of
mathematical objects — and, on the surface, they might not seem very powerful. Noether showed how to exploit such
conditions, however, to maximum advantage: for example, how to use them to show that every set of sub-objects has
a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These
conclusions often are crucial steps in a proof.

Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain
condition, they are called Noetherian in her honor. By definition, a Noetherian ring satisfies an ascending chain
condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly
ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of
submodules breaks off after a finite number. A Noetherian space is a topological space in which every strictly
increasing chain of open subspaces breaks off after a finite number of terms; this definition is made so that the
spectrum of a Noetherian ring is a Noetherian topological space.

The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are
Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and,

Emmy Noether 561

mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of
a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. The chain condition also may be
inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings
are Noetherian, as is the ring of formal power series over a Noetherian ring.

Another application of such chain conditions is in Noetherian induction — also known as well-founded
induction — which is a generalization of mathematical induction. It frequently is used to reduce general statements
about collections of objects to statements about specific objects in that collection. Suppose that S is a partially
ordered set. One way of proving a statement about the objects of S is to assume the existence of a counterexample
and deduce a contradiction, thereby proving the contrapositive of the original statement. The basic premise of
Noetherian induction is that the every non-empty subset of S contains a minimal element. In particular, the set of all
counterexamples contains a minimal element, the minimal counterexample. In order to prove the original statement,
therefore, it suffices to prove something seemingly much weaker: For any counterexample, there is a smaller
counterexample.

Commutative rings, ideals, and modules

T751
Noether's paper, Idealtheorie in Ringbereichen {Theory of Ideals in Ring Domains, 1921), is the foundation of

general commutative ring theory, and gives one of the first general definitions of a commutative ring. Before her

paper, most results in commutative algebra were restricted to special examples of commutative rings, such as

polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the

ascending chain condition on ideals, every ideal is finitely generated. In 1943, French mathematician Claude

Chevalley coined the term, Noetherian ring, to describe this property. A major result in Noether's 1921 paper is

the Lasker— Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of

polynomial rings to all Noetherian rings. The Lasker— Noether theorem can be viewed as a generalization of the

fundamental theorem of arithmetic which states that any positive integer can be expressed as a product of prime

numbers, and that this decomposition is unique.

Noether's work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkorpern (Abstract

T771
Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927) characterized the rings in

which the ideals have unique factorization into prime ideals as the Dedekind domains: integral domains that are

Noetherian, or 1 -dimensional, and integrally closed in their quotient fields. This paper also contains what now are

called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic

results on Noetherian and Artinian modules.

Elimination theory

In 1923—24, Noether applied her ideal theory to elimination theory — in a formulation that she attributed to her
student, Kurt Hentzelt — showing that fundamental theorems about the factorization of polynomials could be carried

T7R1

over directly. Traditionally, elimination theory is concerned with eliminating one or more variables from a system
of polynomial equations, usually by the method of resultants. For illustration, the system of equations often can be
written in the form of a matrix M (missing the variable x) times a vector v (having only different powers of x)
equaling the zero vector, M*v = 0. Hence, the determinant of the matrix M must be zero, providing a new equation in
which the variable x has been eliminated.

Invariant theory of finite groups

Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get

quantitative information about the invariants of a group action, and furthermore, they did not apply to all group

[791
actions. In her 1915 paper, Noether found a solution to the finite basis problem for a finite group of

transformations G acting on a finite dimensional vector space over a field of characteristic zero. Her solution shows

that the ring of invariants is generated by homogenous invariants whose degree is less than, or equal to, the order of

Emmy Noether

562

the finite group; this is called, Noether's bound. Her paper gave two proofs of Noether's bound, both of which also
work when the characteristic of the field is coprime to IGI!, the factorial of the order IGI of the group G. The number
of generators need not satisfy Noether's bound when the characteristic of the field divides the IGI, but Noether
was not able to determine whether the bound was correct when the characteristic of the field divides IGI! but not IGI.
For many years, determining the truth or falsity of the bound in this case was an open problem called "Noether's
gap". It finally was resolved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the
bound remains true.

In her 1926 paper, Noether extended Hilbert's theorem to representations of a finite group over any field; the new
case that did not follow from Hilbert's work, is when the characteristic of the field divides the order of the group.
Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford

ro'}]

conjecture. In this paper Noether also introduced the Noether normalization lemma, showing that a finitely
generated domain A over a field k has a set x , ... , x of algebraically independent elements such that A is integral

over k[x, , ... , x ].

1 n

Contributions to topology

As noted by Pavel Alexandrov and Hermann Weyl in
their obituaries, Noether's contributions to topology
illustrate her generosity with ideas and how her insights
could transform entire fields of mathematics. In
topology, mathematicians study the properties of
objects that remain invariant even under deformation,
properties such as their connectedness. A common joke
is that a topologist cannot distinguish a donut from a
coffee mug, since they can be smoothly deformed into
one another.

Noether is credited with the fundamental ideas that led
to the development of algebraic topology from the
earlier combinatorial topology, specifically, the idea of
homology groups. According to the account of
Alexandrov, Noether attended lectures given by Heinz
Hopf and him in the summers of 1926 and 1927, where
"she continually made observations, which were often

roc]

deep and subtle" and he continues that,

A continuous deformation (homotopy) of a coffee cup into a
doughnut (torus) and back

When ... she first became acquainted with a systematic construction of combinatorial topology, she
immediately observed that it would be worthwhile to study directly the groups of algebraic complexes
and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous
to zero; instead of the usual definition of Betti numbers, she suggested immediately defining the Betti
group as the complementary (quotient) group of the group of all cycles by the subgroup of cycles
homologous to zero. This observation now seems self-evident. But in those years (1925—1928) this was
a completely new point of view

[86]

Noether's suggestion that topology be studied algebraically, was adopted immediately by Hopf, Alexandrov, and
others, and it became a frequent topic of discussion among the mathematicians of Gottingen. Noether
observed that her idea of a Betti group makes the Euler— Poincare formula simple to understand, and Hopf s own

re on T8Q1

work on this subject "bears the imprint of these remarks of Emmy Noether". Noether mentions her own
topology ideas only as an aside in one 1926 publication, where she cites it as an application of group theory.

Emmy Noether

563

The algebraic approach to topology was developed independently in Austria. In a 1926—27 course given in Vienna,
Leopold Vietoris defined a homology group, which was developed by Walther Mayer, into an axiomatic definition in

1928

[92]

Third epoch (1927-35)

Hypercomplex numbers and representation theory

Much work on hypercomplex numbers and group representations was carried out

in the nineteenth and early twentieth centuries, but remained disparate. Noether

united the results and gave the first general representation theory of groups and

[93]
algebras. Briefly, Noether subsumed the structure theory of associative

algebras and the representation theory of groups into a single arithmetic theory of

modules and ideals in rings satisfying ascending chain conditions. This single

work by Noether was of fundamental importance for the development of modern

algebra

[94]

Helmut Hasse worked with Noether

and others to found the theory of

central simple algebras

Noncommutative algebra

Noether also was responsible for a number of other advancements in the field of

algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the

[95]
theory of central simple algebras.

A seminal paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras, which are algebraic
systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a
finite dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is
trivial), and from this, deduced their Hauptsatz ("main theorem"): every finite dimensional central division algebra
over an algebraic number field F splits over a cyclic cyclotomic extension. These theorems allow one to classify all
finite dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a
special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields. This
paper also contains the Skolem— Noether theorem which states that any two embeddings of an extension of a field k
into a finite dimensional central simple algebra over k, are conjugate. The Brauer— Noether theorem
characterization of the splitting fields of a central division algebra over a field.

[98]

gives a

Assessment, recognition, and memorials

Noether's work continues to be relevant for the
development of theoretical physics and mathematics
and she consistently is ranked as one of the greatest
mathematicians of the twentieth century. In his
obituary, fellow algebraist B. L. van der Waerden says

that her mathematical originality was "absolute beyond

[99]
comparison", and Hermann Weyl said that Noether

"changed the face of algebra by her work". During

her lifetime and even until today, Noether has been

characterized as the greatest woman mathematician in

recorded history by mathematicians such as

Pavel Alexandrov, Hermann Weyl, and Jean

Dieudonne.

The Emmy Noether Campus at the University of Siegen is home to
its mathematics and physics departments

Emmy Noether

564

In a letter to The New York Times, Albert Einstein wrote:

In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant
creative mathematical genius thus far produced since the higher education of women began. In the realm
of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered
methods which have proved of enormous importance in the development of the present-day younger
generation of mathematicians.

On 2 January 1935, a few months before her death, mathematician Norbert Wiener wrote that

Miss Noether is ... the greatest woman mathematician who has ever lived; and the greatest woman
scientist of any sort now living, and a scholar at least on the plane of Madame Curie.

At an exhibition at the 1964 World's Fair devoted to Modern Mathematicians, Noether was the only woman
represented among the notable mathematicians of the modern world.

Noether has been honored in several memorials,

• The Association for Women in Mathematics holds a Noether Lecture to honor women in mathematics every year;
in its 2005 pamphlet for the event, the Association characterizes Noether as "one of the great mathematicians of
her time, someone who worked and struggled for what she loved and believed in. Her life and work remain a

tremendous inspiration

„ [106]

Consistent with her dedication to her students, the University of Siegen houses its mathematics and physics
departments in buildings on the Emmy Noether Campus

[107]

• The German Research Foundation (Deutsche Forschungsgemeinschaft) operates the Emmy Noether Programme,

a scholarship providing funding to promising young post-doctorate scholars in their further research and teaching

., ... [108]
activities.

• A street in her hometown, Erlangen, has been named after Emmy Noether and her father, Max Noether.

• The successor to the secondary school she attended in Erlangen has been renamed as the Emmy Noether

School

[103]

In fiction, Emmy Nutter, the physics professor in "The God Patent" by Ransom Stephens, is based on Emmy Noether
[109]

Farther from home,

• The crater Nother on the far side of the Moon is named after her.

• The 7001 Noether asteroid also is named for Emmy Noether.

List of doctoral students

Date

Student name

Dissertation title and English translation

University

Publication

1911.12.16

Falckenberg,
Hans

Verzweigungen von Losungen nichtlinearer
Differentialgleichungen

Ramifications of Solutions of Nonlinear Differential
Equations

Erlangen

Leipzig 1912

1916.03.04

Seidelmann,
Fritz

Die Gesamtheit der kubischen und biquadratischen Gleichungen
mit Affekt bei beliebigem Rationalitatsbereich

Complete Set of Cubic and Biquadratic Equations with Affect
in an Arbitrary Rationality Domain

Erlangen

Erlangen 1916

Emmy Noether

565

1925.02.25

Hermann, Grete

Die Frage der endlich vielen Schritte in der Theorie der
Polynomideale unter Benutzung nachgelassener Satze von Kurt
Hentzelt

The Question of the Finite Number of Steps in the Theory of
Ideals of Polynomials using Theorems of the Late Kurt
Hentzelt §

Gottingen

Berlin 1926

1926.07.14

Grell, Heinrich

Beziehungen zwischen den Idealen verschiedener Ringe
Relationships between the Ideals of Various Rings

Gottingen

Berlin 1927

1927

Dorate,
Wilhelm

fiber einem verallgemeinerten Gruppenbegriff
On a Generalized Conceptions of Groups

Gottingen

Berlin 1927

died before
defense

Holzer, Rudolf

Zur Theorie der primaren Ringe
On the Theory of Primary Rings

Gottingen

Berlin 1927

1929.06.12

Weber, Werner

Idealtheoretische Deutung der Darstellbarkeit beliebiger
natilrlicher Zahlen durch quadratische Formen

Ideal-theoretic Interpretation of the Representability of
Arbitrary Natural Numbers by Quadratic Forms

Gottingen

Berlin 1930

1929.06.26

Levitski, Jakob

fiber vollstandig reduzible Ringe und Unterringe
On Completely Reducible Rings and Subrings

Gottingen

Berlin 1931

1930.06.18

Deuring, Max

Zur arithmetischen Theorie der algebraischen Funktionen
On the Arithmetic Theory of Algebraic Functions

Gottingen

Berlin 1932

1931.07.29

Fitting, Hans

Zur Theorie der Automorphismenringe Abelscher Gruppen und
ihr Analogon bei nichtkommutativen Gruppen

On the Theory of Automorphism-Rings of Abelian Groups and
Their Analogs in Noncommutative Groups

Gottingen

Berlin 1933

1933.07.27

Witt, Ernst

Riemann-Rochscher Satz und Zeta-Funktion im
Hyperkomplexen

The Riemann-Roch Theorem and Zeta Function in
Hypercomplex Numbers

Gottingen

Berlin 1934

1933.12.06

Tsen, Chiungtze

Algebren ilber Funktionenkorper
Algebras over Function Fields'

Gottingen

Gottingen 1934

Emmy Noether

566

1934

Schilling, Otto

fiber gewisse Beziehungen zwischen der Arithmetik
hyperkomplexer Zahlsysteme und algebraischer Zahlkorper

On Certain Relationships between the Arithmetic of
Hypercomplex Number Systems and Algebraic Number
Fields §

Marburg

Braunschweig 1935

1935

Stauffer, Ruth

The construction of a normal basis in a separable extension field

Bryn Mawr

Baltimore 1936

1935

Vorbeck,
Werner

Nichtgaloissche Zerfallungskorper einfacher Systeme
Non-Galois Splitting Fields of Simple Systems

Gottingen

1936

Wichmann,
Wolfgang

Anwendungen der p-adischen Theorie im Nichtkommutativen
Algebren

Applications of the p-adic Theory in Noncommutative
Algebras

Gottingen

Monatshefte fiir Mathematik und
Physik (1936) 44, 203-224.

Eponymous mathematical topics

Noetherian
Noetherian group
Noetherian ring
Noetherian module
Noetherian space

Noetherian induction
Noetherian scheme
Noether normalization lemma
Noether problem

Noether's theorem

Noether's second theorem

Lasker— Noether theorem

Skolem— Noether theorem

Albert— Brauer— Hasse— Noether theorem

Notes

[I] Einstein, Albert. "Professor Einstein Writes in Appreciation of a Fellow- Mathematician", (http://www-history.mcs.st-andrews.ac.uk/
Obits2/Noether_Emmy_Einstein.html) 1 May 1935. Published 5 May 1935 (http://select.nytimes.com/gst/abstract.
html?res=F70DlEFC3D58167A93C6A9178ED85F418385F9) in the New York Times. Online at the MacTutor History of Mathematics
archive. Retrieved on 13 April 2008.

[2] Osen 1974, p. 152; Alexandrov 1981, p. 100.

[3] Ne'eman, Yuval. "The Impact of Emmy Noether's Theorems on XXlst Century Physics", Teicher 1999, p. 83—101.

[4] Weyl 1935

[5] Lederman & Hill 2004, p. 73.

[6] Dick 1981, p. 128

[7] Kimberling 1981, pp. 3-5; Osen 1974, p. 142; Lederman & Hill 2004, pp. 70-71; Dick 1981, pp. 7-9.

[8] Dick 1981, pp. 9-10.

[9] Dick 1981, pp. 10-11; Osen 1974, p. 142.

[10] Dick 1981, pp. 25, 45; Kimberling, p. 5.

[II] Quoted in Kimberling 1981, p. 10.

[12] Dick 1981, pp. 11-12; Kimberling 1981, pp. 8-10; Lederman & Hill 2004, p. 71.

[13] Kimberling 1981, pp. 10—11; Dick 1981, pp. 13—17. Lederman & Hill 2004, p. 71 write that she completed her doctorate at Gottingen, but

this appears to be an error.

[14] Kimberling 1981, pp. 11-12; Dick 1981, pp. 18-24; Osen 1974, p. 143.

[15] Kimberling 1981, p. 14; Dick 1981, p. 32; Osen 1974, pp. 144-145; Lederman & Hill 2004, p. 72.

[16] Dick 1981, pp. 24-26.

[17] Osen 1974, pp. 144-145; Lederman & Hill 2004, p. 72.

[18] Dick 1981, p. 188.

[19] Kimberling 1981, pp. 14-18; Osen 1974, p. 145; Dick 1981, pp. 33-34.

[20] Kimberling 1981, p. 18

[21] Kimberling 1981, p. 18; Dick 1981, pp. 44^15; Osen 1974, pp. 145-146

Emmy Noether

567

[22
[23
[24
[25
[26
[27
[28
[29
[30
[31
[32
[33
[34
[35
[36
[37
[38
[39
[40
[41
[42

[43
[44
[45
[46
[47
[48
[49
[50
[51
[52
[53
[54
[55
[56
[57
[58
[59
[60
[61
[62
[63
[64
[65
[66
[67
[68
[69
[70
[71
[72
[73
[74
[75
[76
[77
[78
[79

van der Waerden 1935, p. 100.

Dick 1981, pp. 57-58; Kimberling 1981, p. 19; Lederman & Hill 2004, p. 74.
Lederman & Hill 2004, p. 74; Osen 1974, p. 148.
Kimberling 1981, pp. 24-25; Dick 1981, pp. 61-63.
Alexandres 1981, pp. 100, 107.
Dick 1981, p. 51.
Dick 1981, pp. 53-57.
Dick 1981, pp. 37-49.
van der Waerden 1935, p. 98.
Dick 1981, pp. 46-48.
Taussky 1981, p. 80
Dick 1981, pp. 40-41.

Scharlau, W. "Emmy Noether's Contributions to the Theory of Algebras" in Teicher 1999, p. 49.
Mac Lane 1981, p. 77; Dick 1981, p. 37.
Dick 1981, pp. 38-41.
Mac Lane 1981, p. 71
Dick 1981, p. 76

Dick 1981, pp. 63-64; Kimberling 1981, p. 26; Alexandrov 1981, pp. 108-110.
Alexandrov 1981, pp. 106-109.
Osen 1974, p. 150; Dick 1981, pp. 82-83.

"Emmy Amalie Noether" (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Noether_Emmy.html). dcs.st-and.ac.uk.
Retrieved 2008-09-04.
Dick 1981, pp. 72-73; Kimberling 1981, pp. 26-27.
Hasse 1933, p. 731

Kimberling 1981, pp. 26-27; Dick 1981, pp. 74-75.
Kimberling 1981, p. 29.

Dick 1981, pp. 75-76; Kimberling 1981, pp. 28-29.
Dick 1981, pp. 78-79; Kimberling 1981, pp. 30-31.
Kimberling 1981, pp. 32-33; Dick 1981, p. 80.
Dick 1981, pp. 80-81.
Dick 1981, pp. 81-82.
Dick 1981, p. 81.

Osen 1974, p. 151; Dick 1981, p. 83.
Dick 1981, p. 82; Kimberling 1981, p. 34.
Kimberling 1981, pp. 37-38.
Kimberling 1981, p. 39.

Osen 1974, pp. 148-149; Kimberling 1981, pp. 11-12.
Gilmer 1981, p. 131.
Kimberling 1981, pp. 10-23.
G.E. Noether 1986, p. 168.
Dicke 1981, p. 101
Noether 1908
M. Noether 1914, p. 11
Gordan 1870
Weyl 1944, pp. 618-621
Hilbert 1890, p. 531
Hilbert 1890, p. 532
Noether 1918.
Noether 1913.
Swan 1969, p. 148.
Malle & Matzat 1999.
Noether 1918b
Kimberling 1981, p. 13.
Lederman & Hill 2004, pp. 97-1 16.
Noether 1921.
Gilmer 1981, p. 133.
Noether 1927.

Noether 1923, Noether 1923b, Noether 1924.
Noether 1915

Emmy Noether

568

[80] Fleischmann 2000, p. 24

[81] Fleischmann 2000, p. 25, Fogarty 2001, p. 5

[82] Noether 1926.

[83] Habousch 1975.

[84] Hilton 1988, p. 284

[85] Dick 1981, p. 173

[86] Dick 1981, p. 174

[87] Hirzebruch, Friedrich. "Emmy Noether and Topology" in Teicher 1999, p. 57—61.

[88] Hopfl928.

[89] Dick 1981, pp. 174-175

[90] Noether 1926b

[91] Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999, p. 63.

[92] Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999, p. 61—63.

[93] Noether 1929.

[94] van der Waerden 1985, p. 244.

[95] Lam 1981, pp. 152-153.

[96] Brauer, Hasse & Noether 1932.

[97] Noether 1933.

[98] Brauer & Noether 1927

[99] Dick 1981, p. 100

[100] James 2002, p. 321.

[101] Dick 1981, p. 154

[102] Dick 1981, p. 152

[103] G.E. Noether 1987, p. 167.

[104] Kimberlingl981,p. 35

[105] Duchin, Moon. "The Sexual Politics of Genius" (http://www.math.lsa.umich.edu/~mduchin/UCD/lll/readings/genius.pdf) (PDF).

December 2004. University of Chicago. Retrieved on 23 March 201 1 (Noether's birthday).
[106] "Introduction" (http://www.awm-math.org/noetherbrochure/Introduction.html). Profiles of Women in Mathematics: The Emmy Noether

Lectures. Association for Women in Mathematics. 2005. Retrieved on 13 April 2008.
[107] "Emmy-Noether-Campus" (http://www.uni-siegen.de/uni/campus/wegweiser/emmy.html). Universitat Siegen. Retrieved on 13 April

2008.
[108] "Emmy Noether Programme: In Brief" (http://www.dfg.de/en/research_funding/programmes/coordinated_programmes/

collaborative_research_centres/modules/emmy_noether/index.html). Research Funding. Deutsche Forschungsgemeinschaft. n.d. Retrieved

on 5 September 2008.
[109] "The God Patent" by Ransom Stephens (http://www.ransomstephens.com/html/literature.html)
[110] Schmadel 2003, p. 570.
[Ill] Blue, Jennifer. Gazetteer of Planetary Nomenclature (http://planetarynames.wr.usgs.gov/). USGS. 25 July 2007. Retrieved on 13 April

2008.

References

Selected works by Emmy Noether (in German)

• Noether, Emmy (1908), "Uber die Bildung des Formensystems der ternaren biquadratischen Form (On Complete
Systems of Invariants for Ternary Biquadratic Forms)" (http://gdz.sub.uni-goettingen.de/no_cache/dms/load/
img/?IDDOC=261200), Journal filr die reine unci angewandte Mathematik 134: 23—90 and two tables.

• Noether, Emmy (1913), "Rationale Funkionenkorper (Rational Function Fields)" (http://gdz.sub.uni-goettingen.
de/no_cache/dms/load/img/?IDDOC=244058), J. Ber. D. DMV22: 316-319.

• Noether, Emmy (1915), "Der Endlichkeitssatz der Invarianten endlicher Gruppen (The Finiteness Theorem for
Invariants of Finite Groups)" (http://www.digizeitschriften.de/index.php?id=loader&
txJkDigiYoohjpilUDDOC]=46H58), Mathematische Annalen 77: 89-92, doi:10.1007/BF01456821

• Noether, Emmy (1918), "Gleichungen mit vorgeschriebener Gruppe (Equations with Prescribed Group)",
Mathematische Annalen 78: 221-229, doi:10.1007/BF01457099.

• Noether, Emmy (1918b), "Invariante Variationsprobleme (Invariant Variation Problems)", Nachr. D. Konig.
Gesellsch. D. Wiss. Zu Gottingen, Math-phys. Klasse 1918: 235—257. English translation by M. A. Tavel (1918),

Emmy Noether 569

arXiv:physics/0503066.

• Noether, Emmy (1921), "Idealtheorie in Ringbereichen (The Theory of Ideals in Ring Domains)" (https://
commerce, metapress.com/content/m3457 w8h62475473/resource-secured/?target=fulltext.pdf&
sid=3y4z44usd0c312452hqwpxm3&sh=www. springerlink.com) (PDF), Mathematische Annalen 83 (1),
ISSN 0025-5831.

• Noether, Emmy; Noether, Emmy (1923), "Zur Theorie der Polynomideale und Resultanten" (http://www.
digizeitschriften.de/index. php?id=loader&tx_jkDigiTools_pil [IDDOC]=362882), Mathematische Annalen 88:
53-79, doi:10.1007/BF01448441.

• Noether, Emmy (1923b), "Eliminationstheorie und allgemeine Idealtheorie" (http://www.digizeitschriften.de/
index. php?id=loader&txJkDigiTools_pil[IDDOC]=362964), Mathematische Annalen 90: 229-261,
doi:10.1007/BF01455443.

• Noether, Emmy (1924), "Eliminationstheorie und Idealtheorie" (http://gdz.sub.uni-goettingen.de/no_cache/
dms/load/img/?IDDOC=248880), Jahresbericht der Deutschen Mathematiker-Vereinigung 33: 1 16— 120.

• Noether, Emmy (1926), "Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p
(Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic p)" (http://gdz.sub.
uni-goettingen.de/no_cache/dms/load/img/?IDDOC=63971), Nachr. Ges. Wiss. Gottingen: 28—35.

• Noether, Emmy (1926b), "Ableitung der Elementarteilertheorie aus der Gruppentheorie (Derivation of the Theory
of Elementary Divisor from Group Theory)" (http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/
?IDDOC=248861), Jahresbericht der Deutschen Mathematiker-Vereinigung 34 (Abt. 2): 104.

• Noether, Emmy (1927), "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkorpern
(Abstract Structure of the Theory of Ideals in Algebraic Number Fields)" (https://commerce.metapress.com/
content/v3t6331n8w244275/resource-secured/?target=fulltext.pdf&sid=zt5psvmwxpvxpbqtyy3riv45&
sh=w ww. springerlink.com) (PDF), Mathematische Annalen 96 (1): 26-61, doi:10.1007/BF01209152.

• Brauer, Richard; Noether, Emmy (1927), "Uber minimale Zerfallungskorper irreduzibler Darstellungen (On the
Minimum Splitting Fields of Irreducible Representations)", Sitz. Ber. D. Preuss. Akad. D. Wiss.: 221—228.

• Noether, Emmy (1929), "Hyperkomplexe Grossen und Darstellungstheorie (Hypercomplex Quantities and the
Theory of Representations) ", Mathematische Annalen 30: 641-692, doi:10.1007/BF01187794.

• Brauer, Richard; Hasse, Helmut; Noether, Emmy (1932), "Beweis eines Hauptsatzes in der Theorie der Algebren
(Proof of a Main Theorem in the Theory of Algebras)" (http://gdz.sub.uni-goettingen.de/no_cache/dms/load/
img/?IDDOC=260847), Journal fur Math. 167: 399-404.

• Noether, Emmy (1933), "Nichtkommutative Algebren (Noncommutative Algebras)", Mathematische Zeitschrift
37: 514-541, doi:10.1007/BF01474591.

• Noether, Emmy (1983), Jacobson, Nathan, ed., Gesammelte Abhandlungen (Collected papers), Berlin-New York:
Springer- Verlag, pp. viii, 777, ISBN 3-540-11504-8, MR0703862.

Additional sources

• Alexandrov, Pavel S. (1981), "In Memory of Emmy Noether", in James W. Brewer and Martha K. Smith, Emmy
Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 99-1 1 1, ISBN 0-8247-1550-0.

• Blue, Meredith (2001) (PDF), Galois Theory and Noether's Problem (http://mccl.mccfl.edu/fl_maa/
proceedings/200 1/blue.pdf), Thirty-Fourth Annual Meeting: Florida Section of The Mathematical Association
of America.

• Nina Byers (1998) " E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation
Laws, (http://arxiv.org/abs/physics/9807044)" in Proceedings of a Symposium on the Heritage of Emmy
Noether, held on 2—4 December 1996, at the Bar-Ilan University, Israel, arXiv:physics/9807044.

• Byers, Nina (2006), "Emmy Noether", in Nina Byers and Gary Williams, Out of the Shadows: Contributions of
20th Century Women to Physics, Cambridge: Cambridge University Press, ISBN 0-521-82197-5.

Emmy Noether 570

Dick, Auguste (1981), Emmy Noether: 1882-1935, Boston: Birkhauser, ISBN 3-7643-3019-8. Trans. H. I.

Blocher.

Fleischmann, Peter (2000), "The Noether bound in invariant theory of finite groups", Advances in Mathematics

156 (1): 23-32, doi:10.1006/aima.2000.1952, MR1800251

Fogarty, John (2001), "On Noether's bound for polynomial invariants of a finite group" (http://www.ams.org/

era/2001-07-02/S1079-6762-01-00088-9/), Electronic Research Announcements of the American Mathematical

Society 7: 5-7, doi:10.1090/S1079-6762-01-00088-9, MR1826990, retrieved 2008-06-16

Gilmer, Robert (1981), "Commutative Ring Theory", in James W. Brewer and Martha K. Smith, Emmy Noether:

A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 131-143, ISBN 0-8247-1550-0.

Gordan, Paul (1870), "Die simultanen Systeme binarer Formen" (in German), Mathematische Annalen 2 (2):

227-280, doi:10.1007/BF01444021.

Haboush, W. J. (1975), "Reductive groups are geometrically reductive" (http://links.jstor.org/

sici?sici=0003-486X(197507)2:102:l<67:RGAGR>2.0.CO;2-l), Ann. Of Math. (The Annals of Mathematics,

Vol. 102, No. 1) 102 (1): 67-83, doi: 10.2307/1970974.

Hasse, Helmut (1933), "Die Struktur der R. Brauerschen Algebrenklassengruppe iiber einem algebraischen

Zahlkorper" (in German), Mathematische Annalen 107: 731-760, doi:10.1007/BF01448916. (German)

Hilbert, David (December 1890), "Ueber die Theorie der algebraischen Formen" (in German), Mathematische

Annalen 36 (4): 473-534, doi:10.1007/BF01208503.

Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century" (http://

www.jstor.org/stable/2689545?origin=JSTOR-pdf), Mathematics Magazine (Mathematics Magazine, Vol. 61,

No. 5) 60 (5): 282-291

Hopf, Heinz (1928), "Eine Verallgemeinerung der Euler-Poincareschen Formel" (http://www.digizeitschriften.

de/index.php?id=loader&txjkDigiTools_pil[IDDOC]=465901) (in German), Nachrichten von der Gesellschaft

der Wissenschaften zu Gottingen. Mathematisch-Physikalische Klasse 2: 127—136.

James, loan (2002), Remarkable Mathematicians from Euler to von Neumann, Cambridge: Cambridge University

Press, ISBN 0-521-81777-3.

Kimberling, Clark (1981), "Emmy Noether and Her Influence", in James W. Brewer and Martha K. Smith, Emmy

Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 3-61, ISBN 0-8247-1550-0.

Lam, Tsit Yuen (1981), "Representation Theory", in James W. Brewer and Martha K. Smith, Emmy Noether: A

Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 145-156, ISBN 0-8247-1550-0.

Lederman, Leon M.; Hill, Christopher T. (2004), Symmetry and the Beautiful Universe, Amherst: Prometheus

Books, ISBN 1-59102-242-8.

Mac Lane, Saunders (1981), "Mathematics at the University of Gottingen 1831—1933", in James W. Brewer and

Martha K. Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 65—78,

ISBN 0-8247-1550-0.

Malle, Gunter; Matzat, Bernd Heinrich (1999), Inverse Galois theory, Springer Monographs in Mathematics,

Berlin, New York: Springer- Verlag, ISBN 978-3-540-62890-3, MR1711577.

Noether, Gottfried E. (1987), Grinstein, L.S. and Campbell, P.J., ed., Women of Mathematics, New York:

Greenwood press, ISBN 0-313-24849-4.

Noether, Max (1914), "Paul Gordan", Mathematische Annalen 75 (1): 1-41, doi:10.1007/BF01564521.

Osen, Lynn M. (1974), "Emmy (Amalie) Noether", Women in Mathematics, MIT Press, pp. 141—152,

ISBN0-262-15014-X.

Schmadel, Lutz D. (2003), Dictionary of Minor Planet Names (5th revised and enlarged ed.), Berlin:

Springer- Verlag, ISBN 3-540-00238-3.

Swan, Richard G. (1969), "Invariant rational functions and a problem of Steenrod", Inventiones Mathematicae 7:

148-158, doi:10.1007/BF01389798.

Emmy Noether 57 1

• Taussky, Olga (1981), "My Personal Recollections of Emmy Noether", in James W. Brewer and Martha K.
Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 79—92,
ISBN 0-8247-1550-0.

• Teicher, M. (ed.) (1999), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Bar-Ilan
University/American Mathematical Society/Oxford University Press, ISBN 978-0-19-851045-1,

OCLC 223099225

• van der Waerden, B.L. (1935), "Nachruf auf Emmy Noether (Obituary of Emmy Noether)", Mathematische
Annalen 111: 469-474, doi:10.1007/BF01472233. Reprinted in Dick 1981. (German)

• van der Waerden, B.L. (1985), A History of Algebra: from al-Khwarizmi to Emmy Noether, Berlin:
Springer- Verlag, ISBN 0-387-13610-X.

• Weyl, Hermann (1935), "Emmy Noether", Scripta Mathematica 3 (3): 201—220, reprinted as an appendix to Dick
(1981).

• Weyl, Hermann (1944), "David Hilbert and his mathematical work", Bulletin of the American Mathematical
Society 50: 612-654, doi:10.1090/S0002-9904-1944-08178-0, MR0011274.

External links

• "Invariante Variationsprobleme", Nachr. v. d. Ges. d. Wiss. zu Gottingen (http://www.physics.ucla.edu/~cwp/
articles/noether.trans/german/emmy235.html) Original paper in German with link to English translation

• "Emmy Noether" in CWP at UCLA (http://cwp.library.ucla.edu/Phase2/
Noether,_Amalie_Emmy @ 86 1 234567 .html)

• Emmy Noether (http://www. genealogy. ams.org/id.php?id=6967) at the Mathematics Genealogy Project

• "Emmy Noether", Biographies of Women Mathematicians (http://www.agnesscott.edu/lriddle/women/
noether.htm), Agnes Scott College

• O'Connor, John J.; Robertson, Edmund F., "Emmy Noether" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Noether_Emmy.html), MacTutor History of Mathematics archive, University of St Andrews.

• Lebenslaufe (http://www.physikerinnen.de/noetherlebenslauf.html) (German) Noether's application for
admission to the University of Erlangen and three curricula vitae, two of which are shown in handwriting, with
transcriptions. The first of these is in Emmy Noether's own handwriting.

• Unpublished (http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=39728) and published (http://gdz.
sub.uni-goettingen.de/dms/load/img/?IDDOC=261200) versions (German) of Noether's 1908 doctoral
dissertation completed at Erlangen.

• Emmy Noether, Mentors & Colleagues (http://faculty.evansville.edu/ck6/bstud/enmc.html) (photo by Clark
Kimberling)

• Oberwolfach collection of photos of Noether (http://owpdb.mfo. de/search?term=noether)

• Correspondence (http://www.univerlag.uni-goettingen.de/hasse-noether/hasse_noether_web.pdf) between
Noether and Helmut Hasse, 1925-35

Norbert Wiener

572

Norbert Wiener

^Sv _ .^TZ-J

mi»B^w^. .

Born

November 26, 1894Columbia, Missouri, U.S.

Died

March 18, 1964 (aged 69)Stockholm, Sweden

Nationality

American

Fields

Mathematics

Cybernetics

Institutions

Massachusetts Institute of Technology

Alma mater

Tufts College BA 1909

Harvard University PhD 1912

Doctoral advisor

Karl Schmidt

Josiah Royce

Doctoral students

Amar Bose

Colin Cherry

Shikao Ikehara

Norman Levinson

Norbert Wiener (November 26, 1894, Columbia, Missouri — March 18, 1964, Stockholm, Sweden) was an
American mathematician.

A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing
work relevant to electronic engineering, electronic communication, and control systems.

Wiener is regarded as the originator of cybernetics, a formalization of the notion of feedback, with many
implications for engineering, systems control, computer science, biology, philosophy, and the organization of
society.

Biography
Youth

Wiener was the first child of Leo Wiener and Bertha Kahn, Jews of Polish and German descent, respectively.
Employing teaching methods of his own invention, Leo educated Norbert at home until 1903, except for a brief
interlude when Norbert was 7 years of age. Wiener became a child prodigy partly due to his father's tutoring. Earning
his living teaching German and Slavic languages, Leo read widely and accumulated a personal library from which
the young Norbert benefited greatly. Leo also had ample ability in mathematics, and tutored his son in the subject
until he left home.

Norbert Wiener 573

After graduating from Ayer High School in 1906 at 1 1 years of age, Wiener entered Tufts College. He was awarded
a BA for mathematics in 1909 at the age of 14, whereupon he began graduate studies of zoology at Harvard. In 1910
he transferred to Cornell to study philosophy.

Harvard and World War I

The next year he returned to Harvard, while still continuing his philosophical studies. Back at Harvard, Wiener
became influenced by Edward Vermilye Huntington, whose mathematical interests ranged from axiomatic
foundations to engineering problems. Harvard awarded Wiener a Ph.D. in 1913, when he was merely 18 years old,
for a dissertation on mathematical logic, supervised by Karl Schmidt, the essential results of which were published as
Wiener (1914). In that dissertation, he was the first to state publicly that ordered pairs can be defined in terms of
elementary set theory. Hence relations can be defined by set theory, thus the theory of relations does not require any
axioms or primitive notions distinct from those of set theory. In 1921, Kazimierz Kuratowski proposed a
simplification of Wiener's definition of ordered pairs, and that simplification has been in common use ever since.

In 1914, Wiener traveled to Europe, to be taught by Bertrand Russell and G. H. Hardy at Cambridge University, and
by David Hilbert and Edmund Landau at the University of Gottingen. During 1915-16, he taught philosophy at
Harvard, then worked as an engineer for General Electric and wrote for the Encyclopedia Americana. Wiener
worked briefly as a journalist for the Boston Herald, where he wrote a feature story on the poor labor conditions for
mill workers in Lawrence, Massachusetts, but he was fired soon afterwards for his reluctance to write favorable
articles about a politician the newspaper's owners sought to promote.

Although Wiener eventually became a staunch pacifist, he eagerly contributed to the war effort in World War I. In
1916, with America's entry into the war drawing closer, Wiener attended a training camp for potential military
officers, but failed to earn a commission. One year later Wiener again tried to join the military, but the government
again rejected him due to his poor eyesight. In the summer of 1918, Oswald Veblen invited Wiener to work on
ballistics at the Aberdeen Proving Ground in Maryland. Living and working with other mathematicians
strengthened his interest in mathematics. However, Wiener was still eager to serve in uniform, and decided to make
one more attempt to enlist, this time as a common soldier. Wiener wrote in a letter to his parents, "I should consider
myself a pretty cheap kind of a swine if I were willing to be an officer but unwilling to be a soldier". This time the
army accepted Wiener into its ranks and assigned him, by coincidence, to a unit stationed at Aberdeen, Maryland.
World War I ended just days after Wiener's return to Aberdeen and Wiener was discharged from the military in
February 1919. [4]

After the war

Wiener was unable to secure a permanent position at Harvard, a situation he blamed largely on anti-semitism at the
university and in particular on the antipathy of Harvard mathematician G D. Birkhoff. He was also rejected for a
position at the University of Melbourne. At W. F. Osgood's suggestion, Wiener became an instructor of mathematics
at MIT, where he spent the remainder of his career, becoming promoted eventually to Professor.

In 1926, Wiener returned to Europe as a Guggenheim scholar. He spent most of his time at Gottingen and with
Hardy at Cambridge, working on Brownian motion, the Fourier integral, Dirichlet's problem, harmonic analysis, and
the Tauberian theorems.

In 1926, Wiener's parents arranged his marriage to a German immigrant, Margaret Engemann; they had two
daughters.

Norbert Wiener 574

During and after World War II

During World War II, his work on the automatic aiming and firing of anti-aircraft guns caused Wiener to study
communication theory and eventually to formulate cybernetics. Unlike many of his contemporaries, Wiener was not
invited to participate in the Manhattan Project. After the war, his fame helped MIT to recruit a research team in
cognitive science, composed of researchers in neuropsychology and the mathematics and biophysics of the nervous
system, including Warren Sturgis McCulloch and Walter Pitts. These men later made pioneering contributions to
computer science and artificial intelligence. Soon after the group was formed, Wiener suddenly ended all contact
with its members, mystifying his colleagues. In their biography of Wiener, Conway and Siegelman suggest that
Wiener's wife Margaret, who detested McCulloch's bohemian lifestyle, engineered the breach.

Wiener later helped develop the theories of cybernetics, robotics, computer control, and automation. He shared his
theories and findings with other researchers, and credited the contributions of others. These included Soviet
researchers and their findings. Wiener's acquaintance with them caused him to be regarded with suspicion during the
"Cold War". He was a strong advocate of automation to improve the standard of living, and to end economic
underdevelopment. His ideas became influential in India, whose government he advised during the 1950s.

After the war, Wiener became increasingly concerned with what he believed was political interference with scientific
research, and the militarization of science. His article "A Scientist Rebels" for the January 1947 issue of The Atlantic

ro]

Monthly urged scientists to consider the ethical implications of their work. After the war, he refused to accept any
government funding or to work on military projects. The way Wiener's beliefs concerning nuclear weapons and the
Cold War contrasted with that of John von Neumann is the major theme of the book John Von Neumann and Norbert
Me«erHeims(1980). [9]

Awards and honors

• Wiener won the Bocher Prize in 1933 and the National Medal of Science in 1963 (Presented by President Johnson
at a White House Ceremony in January 1964.), soon before his death.

• The Norbert Wiener Prize in Applied Mathematics was endowed in 1967 in honor of Norbert Wiener by MIT's
mathematics department and is provided jointly by the American Mathematical Society and Society for Industrial
and Applied Mathematics.

• The Norbert Wiener Award for Social and Professional Responsibility awarded annually by CPSR, was
established in 1987 in honor of Wiener to recognize contributions by computer professionals to socially
responsible use of computers.

• The crater Wiener on the far side of the Moon is named after him.

• The Norbert Wiener Center for Harmonic Analysis and Applications, at the University of Maryland, College
Park, is named in his honor.

• Robert A. Heinlein named a spaceship after him in his 1957 novel Citizen of the Galaxy; a 'Free Trader' ship
called the Norbert Wiener mentioned in Chapter 14.

Work

Information is information, not matter or energy.

— Norbert Wiener, Cybernetics: Or the Control and Communication in the Animal and the Machine

Wiener was an early studier of stochastic and noise processes, contributing work relevant to electronic engineering,
electronic communication, and control systems.

Norbert Wiener

575

Wiener's work with cybernetics influenced Gregory Bateson and Margaret Mead, and through them, Anthropology,
Sociology, and Education.

Wiener equation

A simple mathematical representation of Brownian motion,
the Wiener equation, named after Wiener, assumes the
current velocity of a fluid particle fluctuates.

Wiener filter

For signal processing, the Wiener filter is a filter proposed
by Wiener during the 1940s and published in 1949. Its
purpose is to reduce the amount of noise present in a signal
by comparison with an estimation of the desired noiseless
signal.

In mathematics

z ol

In the mathematical field of probability, the "Wiener sausage"

is a neighborhood of the trace of a Brownian motion up to a

time t, given by taking all points within a fixed distance of

Brownian motion. It can be visualized as a cylinder of fixed

radius the centerline of which is Brownian motion.

Wiener took a great interest in the mathematical theory of

Brownian motion (named after Robert Brown) proving

many results now widely known such as the

non-differentiability of the paths. As a result the

one-dimensional version of Brownian motion became

known as the Wiener process. It is the best known of the Levy processes, cadlag stochastic processes with stationary

statistically independent increments, and occurs frequently in pure and applied mathematics, physics and economics

(e.g. on the stock-market).

Wiener's Tauberian theorem, a 1932 result of Wiener, developed Tauberian theorems in summability theory, on the
face of it a chapter of real analysis, by showing that most of the known results could be encapsulated in a principle
taken from harmonic analysis. As now formulated, the theorem of Wiener does not have any obvious association
with Tauberian theorems, which deal with infinite series; the translation from results formulated for integrals, or
using the language of functional analysis and Banach algebras, is however a relatively routine process.

The Paley— Wiener theorem relates growth properties of entire functions on C and Fourier transformation of
Schwartz distributions of compact support.

The Wiener— Khinchin theorem, (or Wiener — Khintchine theorem or Khinchin — Kolmogorov theorem), states that
the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding
autocorrelation function.

An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent", strictly positive
and locally finite measure on an infinite-dimensional vector space. Wiener's original construction only applied to the
space of real-valued continuous paths on the unit interval, known as classical Wiener space. Leonard Gross provided
the generalization to the case of a general separable Banach space.

The notion of a Banach space itself was discovered independently by both Wiener and Stefan Banach at around the

[12]

same time.

Norbert Wiener 576

Publications

ri3i

Wiener wrote many books and hundreds of articles:

• 1914, A simplification in the logic of relations Proc. Camb. Phil. Soc. vol.13, 387-390 (1912—14). Reprinted in
Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard
Univ. Press: 224-27.

• 1930, Generalized harmonic analysis. Acta Math. 55 1930 117-258.

• 1933, The Fourier Integral and Certain of its Applications Cambridge Univ. Press; reprint by Dover.

• 1942, Extrapolation, Interpolation and Smoothing of Stationary Time Series. A war-time classified report
nicknamed "the yellow peril" because of the color of the cover and the difficulty of the subject. Published postwar
1949 MIT Press, http://www.isss.org/lumwiener.htm])

• 1948, Cybernetics: Or Control and Communication in the Animal and the Machine. Paris, (Hermann & Cie) &
Camb. Mass. (MIT Press) ISBN 9780262730099; 2nd revised ed. 1961.
1950, The Human Use of Human Beings. The Riverside Press (Houghton Mifflin Co.).
1958, Nonlinear Problems in Random Theory. MIT Press & Wiley.
1964, Selected Papers of Norbert Wiener. Cambridge Mass. 1964 (MIT Press & SIAM)
1966, Norbert Wiener 1894-1964. Bull. Amer. Math. Soc. 72(1) 1966. Published in book form.
1966, Generalized Harmonic Analysis and Tauberian Theorems. MIT Press.

1966, God & Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion. MIT Press.
1994, Invention: The Care and Feeding of Ideas. MIT Press.

1976-84, The Mathematical Work of Norbert Wiener. Masani P (ed) 4 vols, Camb. Mass. (MIT Press). This
contains a complete collection of Wiener's mathematical papers with commentaries.

Fiction:

• 1959, The Tempter. Random House.
Autobiography:

• 1953. Ex-Prodigy: My Childhood and Youth. MIT Press.

• 1956. 1 am a Mathematician. London (Gollancz).

Under the name "W. Norbert"

• 1952 The Brain and other short science fiction in Tech Engineering News

References

[I] Conway and Siegelman, 45
[2] Conway and Siegelman, 41-43
[3] Quoted in Conway and Siegelman, 43
[4] Conway and Siegelman, 43-44
[5] Conway and Siegelman, 40, 45
[6] Conway & Siegelman, 127
[7] Conway & Siegelman, 223-227

[8] Norbert Wiener, "A Scientist Rebels," Atlantic Monthly, January, 1947, p. 46.

[9] John Von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, MIT Press; 1980; ISBN 0262081059
[10] Norbert Wiener Center for Harmonic Analysis and Applications (http://www.norbertwiener.umd.edu/), University of Maryland, College

Park

[II] Steve P. Heims, 1977: Gregory Bateson and the mathematicians: From interdisciplinary interaction to societal functions, Journal of the
History of the Behavioral Sciences, Vol. 13. No. 2, pp. 141-159, Wiley Periodicals, Inc., A Wiley Company

[12] Note on a paper ofM. Banach, Fund. Math. 136-143 (1923). See F. Albiac and N. Kalton, Topics in Banach Space Theory (GTM 233). New

York: Springer 2006. p. 15
[13] A full bibliography is given by the Cybernetics Society Publications of Norbert Wiener (http://www.cybsoc.org/wiener.htm)

Norbert Wiener 577

Further reading

• J. M. Almira, 2009. Norbert Wiener. Un matemdtico entre ingenieros (Nobert Wiener. A mathematician between
engineers). Ed. Nivola, Madrid. 420pp. ISBN 978-84-92493-49-4

• Bynum, Terrell W., " Norbert Wiener's Vision: The impact of "the automatic age" on our moral lives, (http://
web.comlab.ox.ac.uk/oucl/research/areas/ieg/e-library/bynum.pdf)"

• Conway, F., and Siegelman, J., 2005. Dark Hero of the Information Age: in search of Norbert Wiener, the father
of cybernetics. Basic Books, New York. 423pp. ISBN 0-7382-0368-8

• Montagnini, Leone, 2005. Le Armonie del disordine (http://www.armoniedeldisordine.it/bookpage.html).
Norbert Wiener Matematico-Filosofo del Novecento. Istituto Veneto di Scienze Lettere edArti, Venezia, 2005.
XVI, 314 pp. ISBN 88-88143-41-6

• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.

• Bluma, Lars, 2005. Norbert Wiener und die Entstehung der Kybernetik im Zweiten Weltkrieg. Mtinster.

• Michel Faucheux, Nobert Wiener, le Golem et la cybernetique, Editions du Sandre,2008

• Heims, Steve J., 1980. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life
and Death. MIT Press.

• Heims, Steve J., 1993. Constructing a Social Science for Postwar America. The Cybernetics Group, 1946-1953.
MIT Press.

• Ilgauds, Hans Joachim, 1980. Norbert Wiener.

• Masani, P. Rustom, 1990. Norbert Wiener 1894-1964. Birkhauser.

A brief profile of Dr. Wiener is given in The Observer newspaper, Sunday, 28 January 1951.

External links

• Norbert Wiener and Cybernetics (http://www.livinginternet.eom/i/ii_wiener.htm) — Living Internet

• O'Connor, John J.; Robertson, Edmund F., "Norbert Wiener" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Wiener_Norbert.html), MacTutor History of Mathematics archive, University of St Andrews.

• Norbert Wiener (http://www. genealogy. ams.org/id.php?id=25222) at the Mathematics Genealogy Project

Paul Dirac

578

Paul Dirac

Paul Adrien Maurice Dirac

Born

Died

Paul Adrien Maurice Dirac8 August 1902Bristol, England
20 October 1984 (aged 82)Tallahassee, Florida, USA

Nationality

Fields

Switzerland (1902-1919)
United Kingdom (1919-1984)

Physics (theoretical)

Institutions

Alma mater

University of Cambridge
Florida State University

University of Bristol
University of Cambridge

Doctoral advisor Ralph Fowler

Doctoral students Homi Bhabha

Harish Chandra Mehta
Dennis Sciama
Behram Kursunoglu
John Polkinghorne

Known for

Notable awards

Dirac equation

Dirac comb

Dirac delta function

Fermi— Dirac statistics

Dirac sea

Dirac spinor

Dirac measure

Bra-ket notation

Dirac adjoint

Dirac large numbers hypothesis

Dirac fermion

Dirac string

Dirac algebra

Dirac operator

Abraham-Lorentz-Dirac force

Dirac bracket

Fermi— Dirac integral

Negative probability

Dirac Picture

Dirac-Coulomb-Breit Equation

Nobel Prize in Physics (1933)
Copley Medal (1952)

Paul Dirac 579

Notes

He is the stepfather of Gabriel Andrew Dirac.

Paul Adrien Maurice Dirac, OM, FRS ( 4) /dl'raek/ di-RAK; 8 August 1902 - 20 October 1984) was an English
theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and
quantum electrodynamics. He held the Lucasian Chair of Mathematics at the University of Cambridge and spent the
last fourteen years of his life at Florida State University.

Among other discoveries, he formulated the Dirac equation, which describes the behaviour of fermions, and
predicted the existence of antimatter.

Dirac shared the Nobel Prize in physics for 1933 with Erwin Schrodinger, "for the discovery of new productive
forms of atomic theory."

Early years

Paul Adrien Maurice Dirac was born at his parents home in Bristol, England on 8 August 1902, and grew up in the
Bishopston area of the city. His father, Charles Adrien Ladislas Dirac, was an immigrant from Saint-Maurice in
the Canton of Valais, Switzerland, who worked in Bristol as a French teacher. His mother, Florence Hannah Dirac
nee Holten, the daughter of a ship's captain, worked as a librarian at the Bristol Central Library. Paul had a younger
sister, Beatrice Isabelle Marguerite, known as Betty, and an older brother, Reginald Charles Felix, known as Felix,

who committed suicide in March 1925. Dirac later recalled: "My parents were terribly distressed. I didn't know
they cared so much. ... I never knew that parents were supposed to care for their children, but from then on I
knew." [7]

ro]

Charles and the children were officially Swiss nationals until they became naturalised on 22 October 1919. Dirac's
father's was strict and authoritarian, although he disapproved of corporal punishment. Dirac had a strained
relationship with his father, so much that after his death, he wrote, "I feel much freer now, and I am my own man."
Charles forced his children to speak to him only in French, in order that they learn the language. When Dirac found
that he could not express what he wanted to say in French, he chose to remain silent.

Dirac was educated first at Bishop Road Primary School, and then at the all-boys Merchant Venturers' Technical
College (later Cotham School), where his father was a French teacher. The school was an institution attached to

ri4i

the University of Bristol, which shared grounds and staff. It emphasised technical subjects like bricklaying,
shoemaking and metal work, and modern languages. This was an unusual arrangement at a time when secondary
education in Britain was still dedicated largely to the classics, and something for which Dirac would later express
gratitude.

Dirac studied Electrical engineering on a City of Bristol University Scholarship at the University of Bristol's
engineering faculty, which was co-located with the Merchant Venturers' Technical College. Shortly before he
completed his degree in 1921, he sat the entrance examination for St John's College, Cambridge. He passed, and was
awarded a £70 scholarship, but this fell short of the amount of money required to live and study at Cambridge.
Despite graduating with a first class honours bachelor of science degree in engineering, the economic climate of the
post-war depression was such that he was unable to find work as an engineer. Instead he took up an offer to study for
bachelor of arts degree in mathematics at the University of Bristol free of charge. He was permitted to skip the first
year of the course owing to his engineering degree.

In 1923, Dirac graduated, once again with first class honours, and received a £140 scholarship from the Department
of Scientific and Industrial Research. Along with his £70 scholarship from St John's College, this was enough to live
at Cambridge. There, Dirac pursued his interests in the theory of general relativity, an interest he gained earlier as a

no]

student in Bristol, and in the nascent field of quantum physics, under the supervision of Ralph Fowler.

Paul Dirac 580

Career

Dirac noticed an analogy between the Poisson brackets of classical mechanics and the recently proposed quantization
rules in Werner Heisenberg's matrix formulation of quantum mechanics. This observation allowed Dirac to obtain
the quantization rules in a novel and more illuminating manner. For this work, published in 1926, he received a
Ph.D. from Cambridge.

In 1928, building on 2x2 spin matrices which he discovered independently (Abraham Pais quoted Dirac as saying "I

ri9i

believe I got these (matrices) independently of Pauli and possibly Pauli got these independently of me" ) of
Wolfgang Pauli's work on non-relativistic spin systems, he proposed the Dirac equation as a relativistic equation of
motion for the wavefunction of the electron. This work led Dirac to predict the existence of the positron, the
electron's antiparticle, which he interpreted in terms of what came to be called the Dirac sea. The positron was
observed by Carl Anderson in 1932. Dirac's equation also contributed to explaining the origin of quantum spin as a
relativistic phenomenon.

The necessity of fermions i.e. matter being created and destroyed in Enrico Fermi's 1934 theory of beta decay,
however, led to a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin h/2,
itself subject to quantization conditions involving anti-commutators. Thus reinterpreted, in 1934 by Werner
Heisenberg, as a (quantum) field equation accurately describing all elementary matter particles- today quarks and
leptons - this Dirac field equation is as central to theoretical physics as the Maxwell, Yang-Mills and Einstein field
equations. Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. He also
introduced the idea of vacuum polarization in the early 1930s. This work was key to the development of quantum
mechanics by the next generation of theorists, and in particular Schwinger, Feynman, Sin-Itiro Tomonaga and Dyson
in their formulation of quantum electrodynamics.

Dirac's Principles of Quantum Mechanics, published in 1930, is a landmark in the history of science. It quickly
became one of the standard textbooks on the subject and is still used today. In that book, Dirac incorporated the
previous work of Werner Heisenberg on matrix mechanics and of Erwin Schrodinger on wave mechanics into a
single mathematical formalism that associates measurable quantities to operators acting on the Hilbert space of

vectors that describe the state of a physical system. The book also introduced the delta function. Following his 1939

T221 T231

article, he also included the bra-ket notation in the third edition of his book, thereby contributing to its

universal use nowadays.

In 1933, following his 1931 paper on magnetic monopoles, Dirac showed that the existence of a single magnetic
monopole in the universe would suffice to explain the observed quantization of electrical charge. In 1975,

ro^i T261 T271 T2R1

1982, and 2009 intriguing results suggested the possible detection of magnetic monopoles, but there is,

to date, no direct evidence for their existence.

Dirac was the Lucasian Professor of Mathematics at Cambridge from 1932 to 1969. In 1937, he proposed a
speculative cosmological model based on the so-called large numbers hypothesis. During World War II, he
conducted important theoretical and experimental research on uranium enrichment by gas centrifuge.

Dirac's quantum electrodynamics made predictions that were - more often than not - infinite and therefore

unacceptable. A workaround known as renormalization was developed, but Dirac never accepted this. "I must say

that I am very dissatisfied with the situation," he said in 1975, "because this so-called 'good theory' does involve

neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible

mathematics. Sensible mathematics involves neglecting a quantity when it is small — not neglecting it just because

[29]
it is infinitely great and you do not want it! His refusal to accept renormalization, resulted in his work on the

subject moving increasingly out of the mainstream. However, from his once rejected notes he managed to work on

putting quantum electrodynamics on "logical foundations" based on Hamiltonian formalism that he formulated. He

found a rather novel way of deriving the anomalous magnetic moment "Schwinger term" and also the Lamb shift,

afresh, using the Heisenberg picture and without using the joining method used by Weisskopf and French, the two

pioneers of modern QED, Schwinger and Feynman, in 1963. That was two years before the

Paul Dirac 581

Tomonaga-Schwinger-Feynman QED was given formal recognition by an award of the Nobel Prize for physics.
Weisskopf and French (FW) were the first to obtain the correct result for the Lamb shift and the anomalous magnetic
moment of the electron. At first FW results did not agree with the incorrect but independent results of Feynman and
Schwinger (Schweber SS 1994 "QED and the men who made it: Dyson,Feynman,Schwinger and Tomonaga",
Princeton :PUP). The 1963-1964 lectures Dirac gave on quantum field theory at Yeshiva University were published
in 1966 as the Belfer Graduate School of Science, Monograph Series Number, 3. After having relocated to Florida in
order to be near his elder daughter, Mary, Dirac spent his last fourteen years (of both life and physics research) at the
University of Miami in Coral Gables, Florida and Florida State University in Tallahassee, Florida.

In the 1950s in his search for a better QED, Paul Dirac developed the Hamiltonian theory of constraints (Canad J
Math 1950 vol 2, 129; 1951 vol 3, 1) based on lectures that he delivered at the 1949 International Mathematical
Congress in Canada. Dirac (1951 "The Hamiltonian Form of Field Dynamics" Canad Jour Math, vol 3 ,1) had also
solved the problem of putting the Tomonaga-Schwinger equation into the Schrodinger representation (See Phillips R
J N 1987 "Tributes to Dirac" p31 London: Adam Hilger) and given explicit expressions for the scalar meson field
(spin zero pion or pseudoscalar meson), the vector meson field (spin one rho meson), and the electromagnetic field
(spin one massless boson, photon).

The Hamiltonian of constrained systems is one of Dirac's many masterpieces. It is a powerful generalization of
Hamiltonian theory that remains valid for curved spacetime. The equations for the Hamiltonian involve only six
degrees of freedom described by Q TS , p rs for each point of the surface on which the state is considered. The g m Q
(m = 0,1,2,3) appear in the theory only through the variables g r0 , (_g 00 ') _1 / 2 which occur as arbitrary
coefficients in the equations of motion. H=J ^ 3 x[ ( — g 00 ) -1 / 2 ^/^ - n T °I a 00 H T ] There are four constraints
or weak equations for each point of the surface x °= constant. Three of them inform the four vector density in the

surface. The fourth Hl is a 3-dimensional scalar density in the surface Hl ~0; H t ~0 (r=l,2,3)

In the late 1950s he applied the Hamiltonian methods he had developed to cast Einstein's general relativity in

Hamiltonian form (Proc Roy Soc 1958,A vol 246, 333,Phys Rev 1959,vol 114, 924) and to bring to a technical

completion the quantization problem of gravitation and bring it also closer to the rest of physics according to Salam

and DeWitt. In 1959 also he gave an invited talk on "Energy of the Gravitational Field" at the New York Meeting of

the American Physical Society later published in 1959 Phys Rev Lett 2, 368. In 1964 he published his "Lectures on

Quantum Mechanics" (London:Academic) which deals with constrained dynamics of nonlinear dynamical systems

including quantization of curved spacetime. He also published a paper entitled "Quantization of the Gravitational

Field" in 1967 ICTP/IAEA Trieste Symposium on Contemporary Physics.

If one considers waves moving in the direction ^resolved into the corresponding Fourier components (r,s = 1,2,3),
the variables in the degrees of freedom 13,23,33 are affected by the changes in the coordinate system whereas those
in the degrees of freedom 12, (11-22) remain invariant under such changes. The expression for the energy splits up
into terms each associated with one of these six degrees of freedom without any cross terms associated with two of
them. The degrees of freedom 13, 23, 33 do not appear at all in the expression for energy of gravitational waves in
the direction x ^. The two degrees of freedom 12, (11-22) contribute a positive definite amount of such a form to
represent the energy of gravitational waves. These two degrees of freedom correspond in the language of quantum
theory , to the gravitational photons (gravitons) with spin +2 or -2 in their direction of motion. The degrees of
freedom (11+22) gives rise to the Newtonian potential energy term showing the gravitational force between the two
positive mass is attractive and the self energy of every mass is negative.
Amongst his many students was John Polkinghorne, who recalls that Dirac "was once asked what was his

fundamental belief. He strode to a blackboard and wrote that the laws of nature should be expressed in beautiful

..[30]
equations.

Paul Dirac 582

Personal life
Family

Dirac married Eugene Wigner's sister, Margit, in 1937. He adopted Margit's two children, Judith and Gabriel. Paul
and Margit Dirac had two children together, both daughters, Mary Elizabeth and Florence Monica.

Margit, known as Manci, visited her brother in 1934 in Princeton from her native Hungary and, while at dinner at the
Annex Restaurant (1930s— 2006 ), met the "lonely-looking man at the next table." This account came from a
physicist from Korea who met and was influenced by Dirac, Y.S. Kim, who has also written: "It is quite fortunate for
the physics community that Manci took good care of our respected Paul A.M. Dirac. Dirac published eleven papers
during the period 1939-46.... Dirac was able to maintain his normal research productivity only because Manci was in
charge of everything else."

A reviewer of the 2009 biography writes: "Dirac blamed his [emotional] frailties on his father, a Swiss immigrant
who bullied his wife, chivvied his children and insisted Paul spoke only French at home, even though the Diracs
lived in Bristol. 'I never knew love or affection when I was a child,' Dirac once said." She also writes that "[t]he
problem lay with his genes. Both father and son had autism, to differing degrees. Hence the Nobel winner's
reticence, literal-mindedness, rigid patterns of behaviour and self-centredness. [Quoting the biography:] 'Dime's
traits as a person with autism were crucial to his success as a theoretical physicist: his ability to order information
about mathematics and physics in a systematic way, his visual imagination, his self-centredness, his concentration
and determination."

Personality

Dirac was known among his colleagues for his precise and taciturn nature. His colleagues in Cambridge jokingly
defined a unit of a dirac which was one word per hour. When Niels Bohr complained that he did not know how to
finish a sentence in a scientific article he was writing, Dirac replied, "I was taught at school never to start a sentence
without knowing the end of it." He criticized the physicist J. Robert Oppenheimer's interest in poetry: "The aim of
science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an
incomprehensible way. The two are incompatible."

Dirac himself wrote in his diary during his postgraduate years that he concentrated solely on his research, and only
stopped on Sunday, when he took long strolls alone.

An anecdote recounted in a review of the 2009 biography tells of Werner Heisenberg and Dirac sailing on a cruise
ship to a conference in Japan in August 1929. "Both still in their twenties, and unmarried, they made an odd couple.
Heisenberg was a ladies' man who constantly flirted and danced, while Dirac — 'an Edwardian geek', as [biographer]
Graham Farmelo puts it — suffered agonies if forced into any kind of socialising or small talk. 'Why do you dance?'

Dirac asked his companion. 'When there are nice girls, it is a pleasure,' Heisenberg replied. Dirac pondered this

T331
notion, then blurted out: 'But, Heisenberg, how do you know beforehand that the girls are nice?"

According to a story told in different versions, a friend or student visited Dirac, not knowing of his marriage.
Noticing the visitor's surprise at seeing an attractive woman in the house, Dirac said, "This is... this is Wigner's
sister". Margit Dirac told both George Gamow and Anton Capri in the 1960s that her husband had actually said,

T371 T3R1

"Allow me to present Wigner's sister, who is now my wife."

Dirac was also noted for his personal modesty. He called the equation for the time evolution of a
quantum-mechanical operator, which he was the first to write down, the "Heisenberg equation of motion". Most
physicists speak of Fermi-Dirac statistics for half-integer-spin particles and Bose-Einstein statistics for integer-spin
particles. While lecturing later in life, Dirac always insisted on calling the former "Fermi statistics". He referred to
the latter as "Einstein statistics" for reasons, he explained, of "symmetry".

Paul Dirac 583

Religious views

Heisenberg recollected a conversation among young participants at the 1927 Solvay Conference about Einstein and
Planck's views on religion. Wolfgang Pauli, Heisenberg and Dirac took part in it. Dirac's contribution was a criticism
of the political purpose of religion, which was much appreciated for its lucidity by Bohr when Heisenberg reported it
to him later. Among other things, Dirac said:

I cannot understand why we idle discussing religion. If we are honest — and scientists have to be — we must admit that religion is a jumble of
false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive
people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear
and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can't for the life of me see
how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why
God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion
is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet.
Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium
that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close
alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards — in heaven if not
on earth — all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the
honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.

[39]

Heisenberg's view was tolerant. Pauli, raised as a Catholic, had kept silent after some initial remarks, but when
finally he was asked for his opinion, said: "Well, our friend Dirac has got a religion and its guiding principle is
'There is no God and Paul Dirac is His prophet.'" Everybody, including Dirac, burst into laughter.

Death and commemoration

In 1984, Dirac died in Tallahassee, Florida and was buried at Tallahassee's Roselawn Cemetery. Dirac's
childhood home in Bristol is commemorated with a blue plaque and the nearby Dirac Road is named in recognition
of his links with the city. A plaque on the wall at the Bishop Road Primary School shows the Dirac equation. A
commemorative stone was erected in a garden Saint-Maurice, Switzerland, the town of origin of his father's family,
on 1 August 1991. On 13 November 1995 a commemorative marker, made from Burlington green slate and inscribed

[41] [431

with the Dirac equation, was unveiled in Westminster Abbey. Objections by the Dean of Westminster,

Edward Carpenter, that Dirac was an atheist were brushed aside.

Dirac shared the 1933 Nobel Prize for physics with Erwin Schrodinger "for the discovery of new productive forms of
atomic theory." Dirac was also awarded the Royal Medal in 1939 and both the Copley Medal and the Max Planck
medal in 1952. He was elected a Fellow of the Royal Society in 1930, an Honorary Fellow of the American Physical
Society in 1948, and an Honorary Fellow of the Institute of Physics, London in 1971. Dirac became a member of the

Order of Merit, an outstanding recognition by the land of his birth, in 1973. He had previously turned down a

1451
knighthood, as he did not want to be addressed by his first name.

In 1975, Dirac gave a series of five lectures at the University of New South Wales which were subsequently
published as a book, Directions of Physics (1978). He donated the royalties from this book to the university for the
establishment of the Dirac Lecture Series. The Silver Dirac Medal for the Advancement of Theoretical Physics is
awarded by the University of New South Wales on the occasion of the lecture.

Immediately after his death, two organisations of professional physicists established annual awards in Dirac's
memory. The Institute of Physics, the United Kingdom's professional body for physicists, awards the Paul Dirac

Medal and Prize for "outstanding contributions to theoretical (including mathematical and computational)

T471
physics". The first three recipients were Stephen Hawking (1987), John Stewart Bell (1988), and Roger Penrose

(1989). The Abdus Salam International Centre for Theoretical Physics (ICTP) awards the Dirac Medal of the ICTP

each year on Dirac's birthday (8 August). Also, the Dirac Prize is awarded by the International Centre for Theoretical

Paul Dirac 584

Physics in his memory. Dirac House in Bristol is the headquarters of Institute of Physics Publishing.

The Dirac-Hellmann Award at Florida State University was endowed by Dr Bruce P. Hellmann (Dirac's last doctoral

student) in 1997 to reward outstanding work in theoretical physics by FSU researchers. The Paul A.M. Dirac

Science Library at Florida State University, which Manci opened in December 1989, is named in his honour, and his

[49]
papers are held there. Outside is a statue of him by Gabriella Bollobas. The street on which the National High

Magnetic Field Laboratory in Tallahassee, Florida, is located was named Paul Dirac Drive. There is also a road

named after him in his home town of Bristol, UK. The BBC named its video codec Dirac in his honour.

Legacy

Dirac is widely regarded as one of the world's greatest physicists. He was one of the founders of quantum mechanics
and quantum electrodynamics.

His early contributions include the modern operator calculus for quantum mechanics, which he called transformation
theory, and an early version of the path integral. He formulated a many-body formalism for quantum mechanics
which allowed each particle to have its own proper time.

His relativistic wave equation for the electron was the first successful attack on the problem of relativistic quantum
mechanics. Dirac founded quantum field theory with his reinterpretation of the Dirac equation as a many-body
equation, which predicted the existence of antimatter and matter— antimatter annihilation. He was the first to
formulate quantum electrodynamics, although he could not calculate arbitrary quantities because the short distance
limit requires renormalization.

In an attempt to solve the quantum divergence problem, Dirac gave a classical point particle theory combining
advanced and retarded waves to eliminate the classical electron self-energy. Although these classical methods did
not immediately solve the problems in quantum electrodynamics, they did lead John Archibald Wheeler and Richard
Feynman to formulate an alternative Green's function description for light, which eventually led to Feynman's point
particle formulation of quantum field theory.

Dirac discovered the magnetic monopole solutions, the first topological configuration in physics, and used them to
give the modern explanation of charge quantization. He developed constrained quantization in the 1960s, identifying
the general quantum rules for arbitrary classical systems.

Dirac's quantum-field analysis of the vibrations of a membrane, in the early 1960s, proved extremely useful to
modern practitioners of superstring theory and its closely related successor, M-Theory.

Bibliography

• Principles of Quantum Mechanics (1930): This book summarizes the ideas of quantum mechanics using the
modern formalism that was largely developed by Dirac himself. Towards the end of the book, he also discusses
the relativistic theory of the electron (the Dirac equation), which was also pioneered by him. This work does not
refer to any other writings then available on quantum mechanics.

• Lectures on Quantum Mechanics (1966): Much of this book deals with quantum mechanics in curved space-time.

• Lectures on Quantum Field Theory (1966): This book lays down the foundations of quantum field theory using
the Hamiltonian formalism.

• Spinors in Hilbert Space (1974): This book based on lectures given in 1969 at the University of Miami, Coral
Gables, Florida, USA, deals with the basic aspects of spinors starting with a real Hilbert space formalism. Dirac
concludes with the prophetic words "We have boson variables appearing automatically in a theory that starts with
only fermion variables, provided the number of fermion variables is infinite. There must be such boson variables
connected with electrons..."

• General Theory of Relativity (1975): This 68-page work summarizes Einstein's general theory of relativity.

Paul Dirac 585

Notes

[I] "The Nobel Prize in Physics 1933" (http://nobelprize.org/nobel_prizes/physics/laureates/1933/). The Nobel Foundation. . Retrieved
2007-11-24.

[2] Farmelo 2009, p. 10
[3] Farmelo 2009, pp. 18-19
[4] Kragh 1990, p. 1
[5] Farmelo 2009, pp. 10-1 1
[6] Farmelo 2009, pp. 77-78
[7] Farmelo 2009, p. 79
[8] Farmelo 2009, p. 34
[9] Farmelo 2009, p. 22
[10] Mehra 1972, p. 17

[II] Kragh 1990, p. 2

[12] Farmelo 2009, pp. 13-17

[13] Farmelo 2009, pp. 20-21

[14] Mehra 1972, p. 18

[15] Farmelo 2009, p. 23

[16] Farmelo 2009, p. 28

[17] Farmelo 2009, pp. 46-47

[18] Farmelo 2009, pp. 52-53

[19] Reminiscences about a great physicist, 1990 ed. Kursunoglu & Wigner, CUP, p. 98

[20] Dirac, P. A. M. (1928-02-01). "The Quantum Theory of the Electron". Proceedings of the Royal Society' of London. Series A, Containing

Papers of a Mathematical and Physical Character 117 (778): 610-624. Bibcode 1928RSPSA.117..610D. doi:10.1098/rspa.l928.0023.
[21] Dirac, Paul A. M. (1933-12-12). "Theory of Electrons and Positrons" (http://nobelprize.org/nobel_prizes/physics/laureates/1933/

dirac-lecture.html). The Nobel Foundation. . Retrieved 2008-11-01.
[22] P. A. M. Dirac (1939). "A New Notation for Quantum Mechanics". Proceedings of the Cambridge Philosophical Society 35 (03): 416.

Bibcode 1939PCPS...35..416D. doi:10.1017/S0305004100021162.
[23] Gieres (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". Reports on Progress in Physics 63 (12): 1893.

arXiv:quant-ph/9907069. Bibcode 2000RPPh...63.1893G. doi: 10. 1088/0034-4885/63/12/201.
[24] P. B. Price; E. K. Shirk; W. Z. Osborne; L. S. Pinsky (1975-08-25). "Evidence for Detection of a Moving Magnetic Monopole". Physical

Review Letters (American Physical Society) 35 (8): 487-490. Bibcode 1975PhRvL..35..487P. doi: 10.1 103/PhysRevLett.35.487.
[25] Bias Cabrera (1982-05-17). "First Results from a Superconductive Detector for Moving Magnetic Monopoles". Physical Review Letters

(American Physical Society) 48 (20): 1378-1381. Bibcode 1982PhRvL..48.1378C. doi:10.1103/PhysRevLett.48.1378.
[26] "Magnetic Monopoles Detected In A Real Magnet For The First Time" (http://www.sciencedaily.com/releases/2009/09/090903163725.

htm). Science Daily. 2009-09-04. . Retrieved 2009-09-04.
[27] D.J.P. Morris, D.A. Tennant, S.A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czter-nasty, M. Meissner, K.C. Rule, J.-U.

Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R.S. Perry (2009-09-03). "Dirac Strings and Magnetic Monopoles in Spin Ice

Dy 2 Ti 2 7 ". Science (Science) 326 (5951): 411. Bibcode 2009Sci...326..411M. doi:10.1126/science.H78868. PMID 19729617.
[28] S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran & T. Fennell (2009-10-15). "Measurement of the charge and current of

magnetic monopoles in spin ice". Nature 461 (7266): 956. Bibcode 2009Natur.461..956B. doi:10.1038/nature08500. PMID 19829376.
[29] Kragh 1990, p. 184

[30] John Polkinghorne. 'Belief in God in an Age of Science' p2
[31] "Last call at The Annex: Nassau Street institution closes doors after more than 70 years" (http://www.dailyprincetonian.com/2006/03/

10/14829/) by Sophia Ahem Dwosh with reporting by Euphemia Mu, The Daily Princetonian, 10 March 2006. Retrieved 2009-02-06.
[32] "Wigner's Sisters" (http://ysfine.com/dirac/wigsis.html) by Y. S. Kim, Department of Physics, University of Maryland, College Park,

Maryland 20742, U.S.A.; written in 1995, article in Web site dedicated to Paul A. M. Dirac. Retrieved 2009-05-08.
[33] "Anti-matter and madness: British physicist Paul Dirac had a brilliant mind, but the joys of daily life flummoxed him" (http://www.

guardian.co.uk/books/2009/feb/01/strangest-man-paul-dirac-review) Review of The Strangest Man by Graham Farmelo by Robin McKie,

The Observer, 1 Feb. 2009. Retrieved 2009-02-06.
[34] Farmelo 2009, p. 89
[35] "Paul Adrien Maurice Dirac" (http://www-history.mcs.st-and.ac.uk/Printonly/Dirac.html). University of St. Andrews. . Retrieved 24

November 2007.
[36] Kragh 1990, p. 258 citing Mehra 1972, pp. 17-59
[37] Gamowl955,p. 121
[38] Capri 2007, p. 148
[39] Heisenberg 1971, pp. 85-86
[40] Heisenberg 1971, p. 87

Paul Dirac 586

[41] this web site "Dirac takes his place next to Isaac Newton" (http://www.fsu.edu/~fstime/FS-Times/Volumel/Issuel/Dirac.html).

Florida State University, this web site. Retrieved 15 April 2011.
[42] "BBC documentary, Everything and Nothing , presented by Professor Jim Al-Khalili , BBC Four, 9:00PM Mon, 28 Mar 2011" (http://

www.bbc.co.uk/iplayer/episode/bOOzwndy/Everything_and_Nothing_Nothing/). BBC. . Retrieved 15 April 2011.
[43] "Paul Dirac" (http://www.dirac.ch/PaulDirac.html). Gisela Dirac. . Retrieved 15 April 2011.
[44] Farmelo 2009, pp. 414-415
[45] Farmelo 2009, pp. 403-404
[46] "Public Dirac Lecture 2008" (http://www.phys.unsw.edu.au/phys_news/Dirac.htm). University of New South Wales. . Retrieved 5

June 2008.
[47] "The Dirac Medal of the Institute of Physics" (http://www.iop.org/activity/awards/Premier_Awards/The_Dirac_Medal_and_Prize/

page_1731.html). Institute of Physics. . Retrieved 24 November 2007.
[48] "Dirac Science Library" (http://www.lib.fsu.edu/about/fsulibraries/dirac/index.html). Florida State University. . Retrieved 15 April

2011.
[49] Farmelo 2009, p. 417

[50] Dirac, P. A. M. (1933). "The Lagrangian in Quantum Mechanics". Physikalische Zeitschrift der Sowjetunion 3: 64—72.
[51] Britain's answer to Einstein (http://www.physics.lsa.umich.edu/nea/news/articles/April02/PH-Duff080702.pdf)

References

• Capri, Anton Z. (2007). Quips, Quotes, and Quanta: An Anecdotal History of Physics (http://books. google.
com/?id=GfmR0mHxeZkC&pg=PA148). Hackensack, New Jersey: World Scientific. ISBN 981-270-919-3.
OCLC 214286147. Retrieved 8 June 2008.

• Crease, Robert P.; Mann, Charles C. (1986). The Second Creation: Makers of the Revolution in Twentieth
Century Physics. New York, New York: Macmillan Publishing. ISBN 0025214403. OCLC 13008048.

• Farmelo, Graham (2009). The Strangest Man: the Life of Paul Dirac. London: Faber and Faber.
ISBN 0465018270. OCLC 426938310.

• Gamow, George (1985). Thirty Years That Shook Physics: The Story of Quantum Theory (http://books. google.
com/?id=L90_wYlVCW0C&pg=PA121). Garden City, New York: Doubleday. ISBN 0-486-24895-X.
OCLC 11970045. Retrieved 8 June 2008.

• Heisenberg, Werner (1971). Physics and Beyond: Encounters and Conversations. New York, New York: Harper
& Row. ISBN 0061316229. OCLC 115992.

• Kragh, Helge (1990). Dirac: A Scientific Biography (http://books. google. com/?id=5ajhJGdL0J4C&
pg=PA184). Cambridge: Cambridge University Press. ISBN 0-521-38089-8-1. OCLC 20013981. Retrieved 8
June 2008.

• Mehra, Jagdish (1972). "The Golden Age of Theoretical Physics: P. A. M. Dirac's Scientific Works from
1924—1933". In Wigner, Eugene Paul; Salam, Abdus. Aspects of Quantum Theory. Cambridge: University Press,
pp. 17-59. ISBN 0521086000. OCLC 532357.

• Schweber, Silvan S. (1994). QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga.
Princeton, New Jersey: Princeton University Press. ISBN 0691036853. OCLC 28966591.

Further reading

• Brown, Helen (24 January 2009). "The Strangest Man: The Hidden Life of Paul Dirac by Graham Farmelo —
review [print version: The man behind the maths]" (http://www.telegraph.co.uk/culture/books/bookreviews/
43 16309/The-Strangest-Man-the-Hidden-Life-of-Paul-Dirac-by-Graham-Farmelo— review.html). The Daily
Telegraph (Review): p. 20. Retrieved 11 April 2011..

• Gilder, Louisa (13 September 2009). "Quantum Leap - Review of The Strangest Man: The Hidden Life of Paul
Dirac by Graham Farmelo'" (http://www.nytimes.com/2009/09/13/books/review/Gilder-t.html?scp=l&
sq=dirac&st=cse). The New York Times (Review). Retrieved 11 April 2011..

Paul Dirac 587

Dirac videos

• Archival footage of Dirac in Princeton 1947 (http://www.youtube. com/watch ?v=PsIlRr65-L4)

• Dirac in 1927 (http://www.youtube.com/watch?v=8GZdZUouzBY)

External links

• Dirac Medal (http://prizes.ictp.it/prizes/Dirac/) of the International Centre for Theoretical Physics

• O'Connor, John J.; Robertson, Edmund F., "Paul Dirac" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Dirac.html), MacTutor History of Mathematics archive, University of St Andrews.

• Dirac Medal (http://www.ch.ic.ac.uk/watoc/) of the World Association of Theoretical and Computational
Chemists (WATOC)

• The Paul Dirac Collection at Florida State University (http://www.lib.fsu.edu/fsulibraries/dirac_collection)

• The Paul A. M. Dirac Collection Finding Aid at Florida State University (http://diglib.lib.fsu.edu/findaids/
dirachtml.html)

• Photocopies of Dirac's papers from the Florida State University collection (http://janus.lib.cam.ac.uk/), held
under Dirac's name in the Archive Centre (http://www.chu.cam.ac.uk/archives/) of Churchill College,
Cambridge, UK

• Letters from Dirac (1932-36) and other papers (http://janus.lib.cam.ac.uk/), held in the Personal Papers
archives (http://www.joh.cam.ac.uk/library/special_collections/personal_papers/) of St John's College,
Cambridge, UK

• Free online access to Dirac's classic papers from Royal Society's Proceedings A (http://royalsocietypublishing.
org/search?submit=yes&submit=yes&submit=Submit&andorexacttitle=and&journalcode=royprsa&
andorexacttitleabs=and&andorexactfulltext=and&authorl=dirac&format=standard&hits=80&
sortspec=relevance& submit=Go)

• Annotated bibliography for Paul Dirac from the Alsos Digital Library for Nuclear Issues (http://alsos.wlu.edu/
qsearch.aspx?browse=people/Dirac,+Paul)

• Oral History interview transcript with Paul Dirac 1 April 1962, 6, 7, 10, & 14 May 1963, American Institute of
Physics, Niels Bohr Library and Archives (http://www.aip.org/history/ohilist/4575_l.html)

• Photos of Paul Dirac at the Emilio Segre Visual Archives, American Institute of Physics (http://photos.aip.org/
veritySearchl.jsp?page=l&chapter=0&collection=storeTest&name=Dirac,+Paul+Adrien+Maurice&
descl=&search=SEARCH+ )

• 2010 June 24 - ScienceTalk Part 1 of interview with Graham Farmelo author of The Strangest Man of Science
(http://www.scientificamerican.com/podcast/episode.cfm?id=the-strangest-man-of-science- 10-06-24)

• 2010 June 24 - ScienceTalk Part 2 of interview with Graham Farmelo author of The Strangest Man of Science
(http://www.scientificamerican.com/podcast/episode.cfm?id=paul-dirac-the-strangest-man-of-sci- 10-06-25)

John von Neumann

588

John von Neumann

Born

John von Neumann in the 1940s
December 28, 1903Budapest, Austria-Hungary

Died
Residence

February 8, 1957 (aged 53)Washington, D.C., United States
United States

Nationality
Fields

Hungarian and American
Mathematics and computer science

Institutions

Alma mater

University of Berlin
Princeton University
Institute for Advanced Study
Site Y, Los Alamos

University of Pazmany Peter
ETH Zurich

Doctoral advisor
Doctoral students

Lipot Fejer

Donald B. Gillies
Israel Halperin
John P. Mayberry

Other notable students Paul Halmos

Clifford Hugh Dowker

John von Neumann

589

Known for

von Neumann Equation

Abelian von Neumann algebra

Duality Theorem

Durbin— Watson statistic

Game theory

von Neumann algebra

von Neumann architecture

Von Neumann bicommutant theorem

Von Neumann cellular automaton

Von Neumann universal constructor

Von Neumann entropy

Von Neumann regular ring

Von Neumann— Bernays— Godel set theory

Von Neumann universe

Von Neumann conjecture

Von Neumann's inequality

Stone— von Neumann theorem

Von Neumann stability analysis

Minimax theorem

Monte Carlo method

Von Neumann extractor

Von Neumann ergodic theorem

Direct integral

Ultrastrong topology

Notable awards

Enrico Fermi Award (1956)
Signature

John von Neumann (English pronunciation: /vDn 'noimen/) (December 28, 1903 — February 8, 1957) was a Hungarian
American mathematician who made major contributions to a vast range of fields, including set theory, functional
analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science,
numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is
generally regarded as one of the greatest mathematicians in modern history. The mathematician Jean Dieudonne
called von Neumann "the last of the great mathematicians", while Peter Lax described him as possessing the most
"fearsome technical prowess" and "scintillating intellect" of the century. Even in Budapest, in the time that
produced geniuses like Theodore von Karman (b. 1881), Leo Szilard (b. 1898), Eugene Wigner (b. 1902), and
Edward Teller (b. 1908), his brilliance stood out.

Von Neumann was a pioneer of the application of operator theory to quantum mechanics, in the development of
functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton
(as one of the few originally appointed), and a key figure in the development of game theory and the concepts

of cellular automata, the universal constructor, and the digital computer. In a short list of facts about his life he
submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on
quantum mechanics, which developed in Gottingen in 1926, and subsequently in Berlin in 1927-1929. Also, my
work on various forms of operator theory, Berlin 1930 and Princeton 1935-1939; on the ergodic theorem, Princeton,
1931-1932." Along with Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics
involved in thermonuclear reactions and the hydrogen bomb.

John von Neumann 590

Biography

The eldest of three brothers, von Neumann was born - Neumann Janos Lajos (Hungarian pronunciation: ['nojniDn
jalnoj lDjoJ]; in Hungarian the family name comes first) on December 28, 1903 in Budapest, Austro-Hungarian
Empire, to wealthy Jewish parents. His father was Neumann Miksa (Max Neumann) who came to Budapest

from Pecs at the end of 1880s, passed doctor of law examinations and worked for a bank. His mother was Kann
Margit (Margaret Kann). [10]

Janos, nicknamed "Jancsi" (Johnny), was a child prodigy, with an aptitude for languages, memorization, and
mathematics. By the age of six, he could exchange jokes in Classical Greek, memorize telephone directories, and
display prodigious mental calculation abilities. He entered the Hungarian-speaking Lutheran high school Fasori
Evangelikus Gimnazium in Budapest in 1911. Although he attended school at the grade level appropriate to his age,
his father hired private tutors to give him advanced instruction in those areas in which he had displayed an aptitude.
Recognized as a mathematical prodigy, he began to study advanced calculus under Gabor Szego at the age of 15. On

ri2i

their first meeting, Szego was so astounded with the boy's mathematical talent that he was brought to tears. In
1913, his father was rewarded with ennoblement for his service to the Austro-Hungarian empire. (After becoming
semi-autonomous in 1867, Hungary had found itself in need of a vibrant mercantile class.) The Neumann family thus
acquiring the title margittai, Neumann Janos became margittai Neumann Janos (John Neumann of Margitta), which
he later changed to the German Johann von Neumann. He received his Ph.D. in mathematics (with minors in
experimental physics and chemistry) from Pazmany Peter University in Budapest at the age of 22. He
simultaneously earned his diploma in chemical engineering from the ETH Zurich in Switzerland at the behest of
his father, who wanted his son to invest his time in a more financially viable endeavour than mathematics. Between

1926 and 1930, he taught as a Privatdozent at the University of Berlin, the youngest in its history. By the end of year

1927 Neumann had published twelve major papers in mathematics, and by the end of year 1929 thirty -two, at a rate
of nearly one major paper per month.

His father, Max von Neumann died in 1929. In 1930, von Neumann, eK,w,K M « MtB , M «*„ m v m *,*,<«>, »>*»-u,» | 8M

Analytische Zahlenlheorle II, Prol. Sctiur, Mo Dl Do Fr 11-12, p. [635

his mother, and his brothers emigrated to the United States. He ^^^ d ^7|."f D^iV2o J d P nialhc ' naUsche L08ih ' Dr " Nei,mann \&&

f • t t • f tii ■ i* ■ ■ ■ Matliematisches Kolloquium, Prof. Bleberbacli, Dr. Fcigl, Prof. Hammerstein,

anglicized his first name to John, keeping the Austrian-aristocratic gr.ihjj, ^Neumann «. «!«,!«» r>™t. E,i,.,asci.midt«ndPro(.scte

surname of von Neumann, whereas his brothers adopted surnames t^^^^K^^^^t^ ,sat«i^ v ,^M^

,j- 1 » t • • i t » r f f • n i Spezielle Funktionen der imthematischen Pliysik, Dr. Neumann von Margitta,

Vonneumann and Neumann (using the de Neumann form briefly when „ m s ° <>-"■ * ^

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Hilbertsche Beweisthcorie, Dr. Neumann von Margitta, Do 16-18, p. [663

Excerpt from the university calendar 1928 of the
Friedrich-Wilhelms-Universitat Berlin

Von Neumann was invited to Princeton University, New Jersey, in
1930, and, subsequently, was one of the first four people selected for

announcing Neumann's lectures on axiomatic set

the faculty of the Institute for Advanced Study (two of the others being theory and logics problems in quantum

Albert Einstein and Kurt Godel), where he remained a mathematics mechanics and special mathematical functions

professor from its formation in 1933 until his death.

In 1937, von Neumann became a naturalized citizen of the U.S. In 1938, he was awarded the Bocher Memorial Prize
for his work in analysis.

John von Neumann

591

Gravestone of John von Neumann

Von Neumann married twice. He married Mariette Kovesi in 1930, just
prior to emigrating to the United States. They had one daughter (von
Neumann's only child), Marina, who is now a distinguished professor
of international trade and public policy at the University of Michigan.
The couple divorced in 1937. In 1938, von Neumann married Klara
Dan, whom he had met during his last trips back to Budapest prior to
the outbreak of World War II. The von Neumanns were very active
socially within the Princeton academic community, and it is from this
aspect of his life that many of the anecdotes which are part of von
Neumann's legend originate.

In 1955, von Neumann was diagnosed with what was either bone or pancreatic cancer. Von Neumann died
a-year-and-a-half later. While at Walter Reed Hospital in Washington, D.C., he invited a Roman Catholic priest,

Father Anselm Strittmatter, O.S.B., to visit him for consultation. This move shocked some of von Neumann's friends

ri4i
in view of his reputation as an agnostic. Von Neumann, however, is reported to have said in explanation that

Pascal had a point, referring to Pascal's wager. Father Strittmatter administered the last sacraments to him. He

died under military security lest he reveal military secrets while heavily medicated. John von Neumann was buried at

Princeton Cemetery in Princeton, Mercer County, New Jersey.

Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied
mathematics. His last work, written while in the hospital and later published in book form as The Computer and the
Brain, gives an indication of the direction of his interests at the time of his death.

Logic and set theory

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigor and breadth
at the end of the 19th century, particularly in arithmetic (thanks to the axiom schema of Richard Dedekind and
Charles Sanders Peirce) and geometry (thanks to David Hilbert). At the beginning of the twentieth century, efforts to
base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not
belong to themselves).

The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (by Ernst
Zermelo and Abraham Fraenkel). Zermelo and Fraenkel provided a series of principles that allowed for the
construction of the sets used in the everyday practice of mathematics: But they did not explicitly exclude the
possibility of the existence of a set that belong to itself. In his doctoral thesis of 1925, von Neumann demonstrated
two techniques to exclude such sets: the axiom of foundation and the notion of class.

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession
of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then
the first must necessarily come before the second in the succession (hence excluding the possibility of a set
belonging to itself.) To demonstrate that the addition of this new axiom to the others did not produce contradictions,
von Neumann introduced a method of demonstration (called the method of inner models) which later became an
essential instrument in set theory.

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to
other classes, while a proper class is defined as a class which does not belong to other classes. Under the
Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to
themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves
can be constructed, but it is a proper class and not a set.

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the
next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer

John von Neumann 592

arrived in September 1930 at the historic mathematical Congress of Konigsberg, in which Kurt Godel announced his
first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove
every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority
of mathematicians of the time. But von Neumann, who had participated at the Congress, confirmed his fame as an
instantaneous thinker, and in less than a month was able to communicate to Godel himself an interesting
consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own
consistency. It is precisely this consequence which has attracted the most attention, even if Godel originally
considered it only a curiosity, and had derived it independently anyway (it is for this reason that the result is called
Godel's second theorem, without mention of von Neumann.)

Quantum mechanics

At the International Congress of Mathematicians of 1900, David Hilbert presented his famous list of twenty-three
problems considered central for the development of the mathematics of the new century. The sixth of these was the
axiomatization of physical theories. Among the new physical theories of the century the only one which had yet to
receive such a treatment by the end of the 1930s was quantum mechanics. Quantum mechanics found itself in a
condition of foundational crisis similar to that of set theory at the beginning of the century, facing problems of both
philosophical and technical natures. On the one hand, its apparent non-determinism had not been reduced to an
explanation of a deterministic form. On the other, there still existed two independent but equivalent heuristic
formulations, the so-called matrix mechanical formulation due to Werner Heisenberg and the wave mechanical
formulation due to Erwin Schrodinger, but there was not yet a single, unified satisfactory theoretical formulation.

After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of
quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a
so-called Hilbert space, analogous to the 6N dimension (N is the number of particles, 3 general coordinate and 3
canonical momentum for each) phase space of classical mechanics but with infinitely many dimensions
(corresponding to the infinitely many possible states of the system) instead: the traditional physical quantities (e.g.,
position and momentum) could therefore be represented as particular linear operators operating in these spaces. The
physics of quantum mechanics was thereby reduced to the mathematics of the linear Hermitian operators on Hilbert
spaces.

For example, the famous uncertainty principle of Heisenberg, according to which the determination of the position of
a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the
two corresponding operators. This new mathematical formulation included as special cases the formulations of both
Heisenberg and Schrodinger, and culminated in the 1932 classic The Mathematical Foundations of Quantum
Mechanics. However, physicists generally ended up preferring another approach to that of von Neumann (which was
considered elegant and satisfactory by mathematicians). This approach was formulated in 1930 by Paul Dirac.

Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs.
non-determinism and in the book he demonstrated a theorem according to which quantum mechanics could not
possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics.
This demonstration contained a conceptual error, but it helped to inaugurate a line of research which, through the
work of John Stuart Bell in 1964 on Bell's Theorem and the experiments of Alain Aspect in 1982, demonstrated that
quantum physics requires a notion of reality substantially different from that of classical physics.

John von Neumann 593

Economics and game theory

Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. Von
Neumann's proved his minimax theorem in 1928. This theorem establishes that in matrix zero-sum games with
perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a
pair of strategies for both players that allows each to minimize his maximum losses (hence the name minimax).
When examining every possible strategy, a player must consider all the possible responses of his adversary. The
player then plays out the strategy which will result in the minimization of his maximum loss. Such strategies, which
minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are
equal (in absolute value) and contrary (in sign).

Von Neumann improved and extended the minimax theorem to include games involving imperfect information and
games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior
(written with Oskar Morgenstern). The public interest in this work was such that The New York Times ran a
front-page story. In this book, von Neumann declared that economic theory needed to use functional analytic
methods, especially convex sets and topological fixed point theorem, rather than the traditional differential calculus,
because the maximum— operator did not preserve differentiable functions. Independently, Leonid Kantorovich's
functional analytic work on mathematical economics also focused attention on optimization theory,
non-differentiability, and vector lattices. Von Neumann's functional-analytic program has dominated economic
theory ever since.

For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using
his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the
matrix pencil A - X B with nonnegative matrices A and B; von Neumann sought probability vectors/? and q and a
positive number X that would solve the complementarity equation

p T (A-XB)q = 0,

along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector
p represents the prices of the goods while the probability vector q represents the "intensity" at which the production
process would run. The unique solution X represents the rate of growth of the economy, which equals the interest
rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were
remarkable achievements, even for von Neumann. Von Neumann's results have been viewed as a special

case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von
Neumann's model of an expanding economy continues to interest mathematical economists with interests in

[22] [231 [241

computational economics. This paper has been called the greatest paper in mathematical economics by

several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary
slackness, and saddlepoint duality.

Building on his results on matrix games and on his model of an expanding economy, Von Neumann also invented
the theory of duality in linear programming, after George B. Dantzig described his work in a few minutes, after an
impatient von Neumann asked him to get to the point. Then, Dantzig listened dumbfounded while von Neumann
provided an hour lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between
matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using
the homogeneous linear system of Gordan (1873) which was later popularized by Karmarkar's algorithm. Von
Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a
nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of
the active simplex). Von Neumann's algorithm was the first interior-point method of linear programming. However,
it was not competitive with the simplex algorithm of Dantzig.

The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored
by the awarding of Nobel prizes in 1972 to Kenneth Arrow, in 1983 to Gerard Debreu, and in 1994 to John Nash
who used fixed point theorems to establish equilibria for noncooperative games and for bargaining problems in his

John von Neumann 594

Ph.D thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans, Leonid
Kantorovich, Wassily Leontief, Paul Samuelson, Robert Dorfman, Robert Solow, and Leonid Hurwicz.

Von Neumann was also the inventor of the method of proof, used in game theory, known as backward induction
(which he first published in 1944 in the book co-authored with Morgenstern, Theory of Games and Economic
Behaviour).

Mathematical statistics and econometrics

Von Neumann made some fundamental contributions to mathematical statistics. In 1941, he derived the exact

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distribution of the ratio of mean square successive difference to the variance for normally distributed variables.

This ratio was applied to the residuals from regression models and is commonly known as the Durbin-Watson

T2R1

statistic for testing the null hypothesis that the errors are serially independent against the alternative that they
follow a stationary first order autoregression. Subsequently, John Denis Sargan and Alok Bhargava extended the
results for testing if the errors on a regression model follow a Gaussian random walk (i.e. possess a unit root) against
the alternative that they are a stationary first order autoregression. Von Neumann's contributions to statistics have
had a major impact on econometric methodology.

Nuclear weapons

Beginning in the late 1930s, von Neumann began to take more of an interest in
applied (as opposed to pure) mathematics. In particular, he developed an
expertise in explosions — phenomena which are difficult to model
mathematically. This led him to a large number of military consultancies,
primarily for the Navy, which in turn led to his involvement in the Manhattan
Project. The involvement included frequent trips by train to the project's secret
research facilities in Los Alamos, New Mexico.

Von Neumann's principal contribution to the atomic bomb itself was in the
concept and design of the explosive lenses needed to compress the plutonium
core of the Trinity test device and the "Fat Man" weapon that was later dropped
on Nagasaki. While von Neumann did not originate the "implosion" concept, he
was one of its most persistent proponents, encouraging its continued
development against the instincts of many of his colleagues, who felt such a
design to be unworkable. The lens shape design work was completed by July
1944.

John von Neumann's wartime Los
Alamos ID badge photo.

In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock
wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave
was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb
would be enhanced with detonation some kilometers above the target, rather than at ground level.

Beginning in the spring of 1945, along with four other scientists and various military personnel, von Neumann was
included in the target selection committee responsible for choosing the Japanese cities of Hiroshima and Nagasaki as
the first targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb
blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum
shock wave propagation and thus maximum effect. The cultural capital Kyoto, which had been spared the
firebombing inflicted upon militarily significant target cities like Tokyo in World War II, was von Neumann's first

choice, a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed

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by Secretary of War Henry Stimson. 1

John von Neumann 595

On July 16, 1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the first atomic
bomb blast, conducted as a test of the implosion method device, 35 miles (56 km) southeast of Socorro, New
Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5
kilotons of TNT, but Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of torn-up
paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion
had been between 20 and 22 kilotons.

After the war, Robert Oppenheimer remarked that the physicists involved in the Manhattan project had "known sin".
Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."

Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained
the hydrogen bomb project. He then collaborated with Klaus Fuchs on further development of the bomb, and in 1946

the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined

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a scheme for using a fission bomb to compress fusion fuel to initiate a thermonuclear reaction. The Fuchs-von

Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen

bomb design, the Teller-Ulam design. Their work was, however, incorporated into the "George" shot of Operation

Greenhouse, which was instructive in testing out concepts that went into the final design. The Fuchs-von Neumann

work was passed on, by Fuchs, to the USSR as part of his nuclear espionage, but it was not used in the Soviet's own,

independent development of the Teller-Ulam design. The historian Jeremy Bernstein has pointed out that ironically,

"John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole

course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been

successfully made."

Computer science

Von Neumann's hydrogen bomb work was also played out in the realm of computing, where he and Stanislaw Ulam
developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he
contributed to the development of the Monte Carlo method, which allowed complicated problems to be
approximated using random numbers. Because using lists of "truly" random numbers was extremely slow, von
Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this
method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other
method at his disposal, and also noted that when it went awry it did so obviously, unlike methods which could be
subtly incorrect.

While consulting for the Moore School of Electrical Engineering at the University of Pennsylvania on the ED VAC
project, von Neumann wrote an incomplete First Draft of a Report on the EDVAC. The paper, which was widely
distributed, described a computer architecture in which the data and the program are both stored in the computer's
memory in the same address space. This architecture is to this day the basis of modern computer design, unlike the
earliest computers that were 'programmed' by altering the electronic circuitry. Although the single-memory, stored
program architecture is commonly called von Neumann architecture as a result of von Neumann's paper, the
architecture's description was based partly on the work of J. Presper Eckert and John William Mauchly, inventors
of the ENIAC at the University of Pennsylvania.

Von Neumann also created the field of cellular automata without the aid of computers, constructing the first

self-replicating automata with pencil and graph paper. The concept of a universal constructor was fleshed out in his

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posthumous work Theory of Self Reproducing Automata. Von Neumann proved that the most effective way of

performing large-scale mining operations such as mining an entire moon or asteroid belt would be by using

self-replicating machines, taking advantage of their exponential growth.

He is credited with at least one contribution to the study of algorithms. Donald Knuth cites von Neumann as the
inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted

hoi T3Q1

recursively and then merged together. His algorithm for simulating a fair coin with a biased coin is used in the

John von Neumann 596

"software whitening" stage of some hardware random number generators.

He also engaged in exploration of problems in numerical hydrodynamics. With R. D. Richtmyer he developed an
algorithm defining artificial viscosity that improved the understanding of shock waves. It is possible that we would
not understand much of astrophysics, and might not have highly developed jet and rocket engines without that work.
The problem was that when computers solve hydrodynamic or aerodynamic problems, they try to put too many
computational grid points at regions of sharp discontinuity (shock waves). The artificial viscosity was a
mathematical trick to slightly smooth the shock transition without sacrificing basic physics.

Politics and social affairs

Von Neumann obtained at the age of 29 one of the first five professorships at the new Institute for Advanced Study
in Princeton, New Jersey (another had gone to Albert Einstein). He was a frequent consultant for the Central
Intelligence Agency, the United States Army, the RAND Corporation, Standard Oil, IBM, and others.

Throughout his life von Neumann had a respect and admiration for business and government leaders; something
which was often at variance with the inclinations of his scientific colleagues. Von Neumann entered government
service (Manhattan Project) primarily because he felt that, if freedom and civilization were to survive, it would have

to be because the U.S. would triumph over totalitarianism from the right (Nazism and Fascism) and totalitarianism

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from the left (Soviet Communism).

As President of the Von Neumann Committee for Missiles, and later as a member of the United States Atomic
Energy Commission, from 1953 until his death in 1957, he was influential in setting U.S. scientific and military
policy. Through his committee, he developed various scenarios of nuclear proliferation, the development of
intercontinental and submarine missiles with atomic warheads, and the controversial strategic equilibrium called
mutual assured destruction. During a Senate committee hearing he described his political ideology as "violently
anti-communist, and much more militaristic than the norm".

Von Neumann's interest in meteorological prediction led him to propose manipulating the environment by spreading
colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedo), thereby raising
global temperatures. He also favored a preemptive nuclear attack on the Soviet Union, believing that doing so could
prevent it from obtaining the atomic bomb.

Personality

Von Neumann invariably wore a conservative grey flannel business suit, once riding down the Grand Canyon astride

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a mule in a three-piece pin-stripe, and he enjoyed throwing large parties at his home in Princeton, occasionally

T421 [431

twice a week. His white clapboard house at 26 Westcott Road was one of the largest in Princeton. Despite
being a notoriously bad driver, he nonetheless enjoyed driving (frequently while reading a book) — occasioning
numerous arrests as well as accidents. When Cuthbert Hurd hired him as a consultant to IBM, Hurd often quietly
paid the fines for his traffic tickets.

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed
Yiddish and "off-color" humor (especially limericks).

John von Neumann 597

Honors

• The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences
(INFORMS, previously TIMS-ORSA) is awarded annually to an individual (or group) who have made
fundamental and sustained contributions to theory in operations research and the management sciences.

• The IEEE John von Neumann Medal is awarded annually by the IEEE "for outstanding achievements in
computer-related science and technology."

• The John von Neumann Lecture is given annually at the Society for Industrial and Applied Mathematics (SIAM)
by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary
prize.

• The crater Von Neumann on the Moon is named after him.

• The John von Neumann Computing Center in Princeton, New Jersey (40°20'55"N 74°35'32"W) was named in
his honour.

• The professional society of Hungarian computer scientists, John von Neumann Computer Society, is named after
John von Neumann.

• On February 15, 1956, Neumann was presented with the Presidential Medal of Freedom by President Dwight
Eisenhower.

• On May 4, 2005 the United States Postal Service issued the American Scientists commemorative postage stamp
series, a set of four 37-cent self-adhesive stamps in several configurations. The scientists depicted were John von
Neumann, Barbara McClintock, Josiah Willard Gibbs, and Richard Feynman.

• The John von Neumann Award of the Rajk Laszlo College for Advanced Studies was named in his honour, and
has been given every year since 1995 to professors who have made an outstanding contribution to the exact social
sciences and through their work have strongly influenced the professional development and thinking of the
members of the college.

Selected works

• 1923. On the introduction of transfinite numbers, 346-54.

• 1925. An axiomatization of set theory, 393—413.

• 1932. Mathematical Foundations of Quantum Mechanics, Beyer, R. T., trans., Princeton Univ. Press. 1996
edition: ISBN 0-691-02893-1.

• 1944. Theory of Games and Economic Behavior, with Morgenstern, O., Princeton Univ. Press. 2007 edition:
ISBN 978-0-691-13061-3.

• 1 945 . First Draft of a Report on the EDVA C TheFirstDraft.pdf [46]

• 1963. Collected Works of John von Neumann, Taub, A. H., ed., Pergamon Press. ISBN 0080095666

• 1966. Theory of Self-Reproducing Automata, Burks, A. W., ed., Univ. of Illinois Press.

Biographical material

• Aspray, William, 1990. John von Neumann and the Origins of Modern Computing.

• Chiara, Dalla, Maria Luisa and Giuntini, Roberto 1997, La Logica Quantistica in Boniolo, Giovani, ed., Filosofia
della Fisica (Philosophy of Physics). Bruno Mondadori.

• Goldstine, Herman, 1980. The Computer from Pascal to von Neumann.

• Halmos, Paul R., 1985. / Want To Be A Mathematician Springer- Verlag

• Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903—1957). Teil 1: Lehrjahre eines judischen
Mathematikers wahrend der Zeit der Weimarer Republik. In: Informatik-Spektrum 29 (2), S. 133—141.

• Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903—1957). Teil 2: Ein Privatdozent auf dem Weg
von Berlin nach Princeton. In: Informatik-Spektrum 29 (3), S. 227—236.

John von Neumann 598

• Heims, Steve J., 1980. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life
and Death MIT Press

• Macrae, Norman, 1999. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game
Theory, Nuclear Deterrence, and Much More. Reprinted by the American Mathematical Society.

• Poundstone, William. Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. 1992.

• Redei, Miklos (ed.), 2005 John von Neumann: Selected Letters American Mathematical Society

• Ulam, Stanislaw, 1983. Adventures of a Mathematician Scribner's

• Vonneuman, Nicholas A. John von Neumann as Seen by His Brother ISBN 0-9619681-0-9

• 1958, Bulletin of the American Mathematical Society 64.

• 1990. Proceedings of the American Mathematical Society Symposia in Pure Mathematics 50.

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• John von Neumann 1903—1957 , biographical memoir by S. Bochner, National Academy of Sciences, 1958

Popular periodicals

• Good Housekeeping Magazine, September 1956 Married to a Man Who Believes the Mind Can Move the World

• Life Magazine, February 25, 1957 Passing of a Great Mind

Video

• John von Neumann, A Documentary (60 min.), Mathematical Association of America

Notes

[I] Ed Regis (1992-11-08). "Johnny Jiggles the Planet" (http://query.nytimes.com/gst/fullpage.
html?res=9E0CE7D91239F93BA35752ClA964958260). The New York Times. . Retrieved 2008-02-04.

[2] Impagliazzo, p. vii

[3] Dictionary of Scientific Bibliography, ed. C. C. Gillispie, Scibners, 1981

[4] Impagliazzo, p. 7

[5] Doran, p. 2

[6] Nelson, David (2003). The Penguin Dictionary of Mathematics. London: Penguin, pp. 178-179. ISBN 0-141-01077-0.

[7] Doran, p. 1

[8] Nathan Myhrvold, "John von Neumann", (http://www.time.eom/time/magazine/article/0, 9171, 21839, 00.html) Time, March 21, 1999.

Accessed September 5, 2010
[9] Clay Blair, Jr. "Passing of a Great Mind". (http://books. google.com/books ?id=rEEEAAAAMBAJ&pg=PA104&dq="John+von+

NeumannaDD"+parents+jewish&hl=en&ei=5KSDTMHGJIaHOMrE9M0O&sa=X&oi=book_result&ct=result&resnum=8&

ved=0CEkQ6AEwBw#v=onepage&q="John von NeumannaOD" &f=false) Life, February 25, 1957; p. 104
[10] Norman Macrae (June 2000). John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear

Deterrence, and Much More (http://books. google.com/books ?id=00-_gSRhe-EC&pg=PA37). AMS Bookstore, pp. 37-38.

ISBN 978082 1 826768. . Retrieved 24 March 201 1 .

[II] William Poundstone, Prisoner's dilemma (Oxford, 1993), introduction
[12] Impagliazzo, p. 5

[13] While there is a general agreement that the initially discovered bone tumor was a secondary growth, sources differ as to the location of the

primary cancer. While Macrae gives it as pancreatic, the Life magazine article says it was prostate.
[14] The question of whether or not von Neumann had formally converted to Catholicism upon his marriage to Mariette Kovesi (who was

Catholic) is addressed in Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973),

pp. 382—394. He was baptised Roman Catholic, but certainly was not a practicing member of that religion after his divorce.
[15] Marion Ledwig. "The Rationality of Faith" (http://sammelpunkt.philo.at:8080/1647/l/ledwig.pdf), citing Macrae, p. 379.
[16] Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382-394
[17] John von Neumann at Find a Grave (http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=7333144)
[18] For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing

that of the Perron— Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem

A - X I q = 0,

where the nonnegative matrix A must be square and where the diagonal matrix / is the identity matrix. Von Neumann's irreducibility condition
was called the "whales and wranglers" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English
translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every
economic good. Weaker "irreducibility" conditions were given by David Gale and by John Kemeny, Oskar Morgenstern, and Gerald L.
Thompson in the 1950s and then by Stephen M. Robinson in the 1970s.
[19] David Gale. The theory of linear economic models. McGraw-Hill, New York, 1960.

John von Neumann 599

[20] Morgenstern, Oskar; Thompson, Gerald L. (1976). Mathematical theory of expanding and contracting economies. Lexington Books.

Lexington, Massachusetts: D. C. Heath and Company, pp. xviii+277. ISBN 0669000892.
[21] Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0-471-98232-6.

• Rockafellar, R. Tyrrell. Monotone processes of convex and concave type. Memoirs of the American Mathematical Society. Providence,
R.I.: American Mathematical Society, pp. i+74. ISBN 0821812777.

• Rockafellar, R. T. (1974). "Convex algebra and duality in dynamic models of production". In Josef Loz and Maria Loz. Mathematical
models in economics (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972). Amsterdam: North-Holland and Polish Adademy of
Sciences (PAN), pp. 351-378.

• Rockafellar, R. T. (1970 (Reprint 1997 as a Princeton classic in mathematics)). Convex analysis. Princeton, NJ: Princeton University
Press. ISBN 0691080690.

[23] Kenneth Arrow, Paul Samuelson, John Harsanyi, Sidney Afriat, Gerald L. Thompson, and Nicholas Kaldor. (1989). Mohammed Dore,

Sukhamoy Chakravarty, Richard Goodwin, ed. John Von Neumann and modern economics. Oxford:Clarendon. pp. 261.
[24] Chapter 9.1 "The von Neumann growth model" (pages 277—299): Yinyu Ye. Interior point algorithms: Theory and analysis. Wiley. 1997.
[25] George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. Springer- Verlag.
[26] John MacQuarrie. "Mathematics and Chess" (http://www-groups.dcs.st-and.ac.uk/~history/Projects/MacQuarrie/Chapters/Ch4.html).

School of Mathematics and Statistics, University of St Andrews, Scotland. . Retrieved 2007-10- 18. "Others claim he used a method of proof,

known as 'backwards induction' that was not employed until 1953, by von Neumann and Morgenstern. Ken Binmore (1992) writes, Zermelo

used this method way back in 1912 to analyze Chess. It requires starting from the end of the game and then working backwards to its

beginning, (p. 32)"
[27] von Neumann, John. (1941). "Distribution of the ratio of the mean square successive difference to the variance" (http://projecteuclid.org/

DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177731677). Annals of Mathematical Statistics, 12, 367—395. (

JSTOR (http://www.jstor.org/pss/223595 1))
[28] Durbin, J., and Watson, G. S. (1950) "Testing for Serial Correlation in Least Squares Regression, I." (http://www.jstor.org/pss/2332391)

Biometrika 37, 409-428.
[29] Sargan, J.D. and Alok Bhargava (1983). "Testing residuals from least squares regression for being generated by the Gaussian random walk"

(http://www.jstor.org/pss/1912252). Econometrica, 51, p. 153-174.
[30] Lillian Hoddeson ... . With contributions from Gordon Baym ...; "Lillian Hoddeson, Paul W. Henriksen, Roger A. Meade, Catherine

Westfall (1993). Critical Assembly: A Technical History of Los Alamos during the Oppenheimer Years, 1943—1945. Cambridge, UK:

Cambridge University Press. ISBN 0-521-44132-3.
[31] Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Touchstone Simon & Schuster. ISBN 0-684-81378-5.
[32] Groves, Leslie (1962). Now It Can Be Told: The Story of the Manhattan Project. New York: Da Capo. ISBN 0-306-80189-2.
[33] Herken, pp. 171, 374
[34] Bernstein, Jeremy (2010). "John von Neumann and Klaus Fuchs: an Unlikely Collaboration". Physics in Perspective 12: 36.

doi:10.1007/s00016-009-0001-l.
[35] "Conversation with Marina Whitman" (http://256.com/gray/docs/misc/conversation_with_marina_whitman.shtml). Gray Watson

(256.com). . Retrieved 2011-01-30.
[36] The name for the architecture is discussed in John W. Mauchly and the Development of the ENIAC Computer (http://www. library. upenn.

edu/exhibits/rbm/mauchly/jwm9.html), part of the online ENIAC museum (http://www.seas.upenn.edu/~museum/), in Robert Slater's

computer history book, Portraits in Silicon, and in Nancy Stern's book From ENIAC to UNIVAC.
[37] John von Neumann (1966). Arthur W. Burks, ed. Theory: of Self-Reproducing Automata. Urbana and London: Univ. of Illinois Press.

ISBN 0598377980. PDF reprint (http://www.history-computer.com/Library/VonNeumannl.pdf)
[38] Knuth, Donald (1998). The Art of Computer Programming: Volume 3 Sorting and Searching. Boston: Addison- Wesley, pp. 159.

ISBN 0-201-89685-0.
[39] von Neumann, John (1951). "Various techniques used in connection with random digits". National Bureau of Standards Applied Math Series

12: 36.
[40] Mathematical Association of American documentary, especially comments by Morgenstern regarding this aspect of von Neumann's

personality
[41] Macrae, p. 332; Heims, pp. 236-247.
[42] Macrae, pp. 170-171

[43] Ed Regis. Who Got Einstein's Office?: Eccentricity and Genius at the Institute for Advanced Study. Perseus Books 1988 p 103
[44] Nancy Stern (January 20, 1981). "An Interview with Cuthbert C. Hurd" (http://www.cbi. umn.edu/oh/pdf.phtml?id=159). Charles

Babbage Institute, University of Minnesota. . Retrieved June 3, 2010.
[45] "Introducing the John von Neumann Computer Society" (http://www. njszt.hu/neumann/neumann. head.page?nodeid=210). John von

Neumann Computer Society. . Retrieved 2008-05-20.
[46] http://systemcomputing.org/turing%20award/Maurice_1967/TheFirstDraft.pdf
[47] http://books.nap.edu/html/biomems/jvonneumann.pdf

John von Neumann 600

References

This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed
under the GFDL.

• Doran, Robert S.; John Von Neumann, Marshall Harvey Stone, Richard V. Kadison, American Mathematical
Society (2004). Operator Algebras, Quantization, and None ommutative Geometry: A Centennial Celebration
Honoring John Von Neumann and Marshall H. Stone (http://books. google. com/?id=m5bSoD9XsfoC&
pg=PAl). American Mathematical Society Bookstore. ISBN 9780821834022.

• Heims, Steve J. (1980). John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life
and Death. Cambridge, Massachusetts: MIT Press. ISBN 0262081059.

• Herken, Gregg (2002). Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer,
Ernest Lawrence, and Edward Teller. ISBN 978-0805065886.

• Impagliazzo, John; Glimm, James; Singer, Isadore Manuel The Legacy of John von Neumann (http://books.
google.com/books?id=XBK-r0gS0YMC&pg=PA15), American Mathematical Society 1990 ISBN 0821842196

• Israel, Giorgio; Ana Millan Gasca (1995). The World as a Mathematical Game: John von Neumann, Twentieth
Century Scientist.

• Macrae, Norman (1992). John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game
Theory, Nuclear Deterrence, and Much More. Pantheon Press. ISBN 0679413081.

• Slater, Robert (1989). Portraits in Silicon. Cambridge, Mass.: MIT Press, pp. 23-33. ISBN 0262691310.

External links

• O'Connor, John J.; Robertson, Edmund F., "John von Neumann" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Von_Neumann.html), MacTutor History of Mathematics archive, University of St Andrews.

• von Neumann's contribution to economics (http://www.findarticles.eom/p/articles/mi_mOIMR/is_3-4_79/
ai_l 13 139424) — International Social Science Review

• Oral history interview with Alice R. Burks and Arthur W. Burks (http://www.cbi.umn.edu/oh/display.
phtml?id=43), Charles Babbage Institute, University of Minnesota, Minneapolis. Alice Burks and Arthur Burks
describe ENIAC, ED VAC, and IAS computers, and John von Neumann's contribution to the development of
computers.

• Oral history interview with Eugene P. Wigner (http://www.cbi. umn.edu/oh/display.phtml?id=77), Charles
Babbage Institute, University of Minnesota, Minneapolis. Wigner talks about his association with John von
Neumann during their school years in Hungary, their graduate studies in Berlin, and their appointments to
Princeton in 1930. Wigner discusses von Neumann's contributions to the theory of quantum mechanics, and von
Neumann's interest in the application of theory to the atomic bomb project.

• Oral history interview with Nicholas C. Metropolis (http://www.cbi.umn.edu/oh/display.phtml?id=81),
Charles Babbage Institute, University of Minnesota. Metropolis, the first director of computing services at Los
Alamos National Laboratory, discusses John von Neumann's work in computing. Most of the interview concerns
activity at Los Alamos: how von Neumann came to consult at the laboratory; his scientific contacts there,
including Metropolis; von Neumann's first hands-on experience with punched card equipment; his contributions
to shock-fitting and the implosion problem; interactions between, and comparisons of von Neumann and Enrico
Fermi; and the development of Monte Carlo methods. Other topics include: the relationship between Alan Turing
and von Neumann; work on numerical methods for non-linear problems; and the ENIAC calculations done for
Los Alamos.

• Von Neumann vs. Dirac (http://plato.stanford.edu/entries/qt-nvd/) — from Stanford Encyclopedia of
Philosophy.

• John von Neumann Postdoctoral Fellowship — Sandia National Laboratories (http://www.sandia.gov/careers/
fellowships. html#jon)

John von Neumann 601

• Von Neumann's Universe (http://www.itconversations.com/shows/detail454.html), audio talk by George
Dyson

• John von Neumann's 100th Birthday (http://www.stephenwolfram.com/publications/recent/neumann/),
article by Stephen Wolfram on Neumann's 100th birthday.

• Annotated bibliography for John von Neumann from the Alsos Digital Library for Nuclear Issues (http://alsos.
wlu.edu/qsearch.aspx?browse=people/Neumann,+John+von)

• Budapest Tech Polytechnical Institution — John von Neumann Faculty of Informatics (http://nik.bmf.hu/)

• John von Neumann speaking at the dedication of the NORD (http://elm.eeng.dcu.ie/~alife/
von-neumann-1954-NORD/), December 2, 1954 (audio recording)

• The American Presidency Project (http://www. presidency. ucsb.edu/ws/index. php?pid=10735)

• John Von Neumann Memorial (http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=7333144) at Find
A Grave

George Birkhoff

602

George Birkhoff

George David Birkhoff

Born

21 March 18840verisel, Michigan

Died

12 November 1944 (aged 60)Cambridge

Massachusetts

Nationality

HB American

Fields

Mathematics

Institutions

Harvard University
Yale University
Princeton University
Radcliffe College

Alma mater

University of Chicago

Doctoral advisor

E. H. Moore

Doctoral students

David Bourgin
Robert D. Carmichael
Hyman Ettlinger
Bernard Koopman
Rudolph Langer
Marston Morse
Marshall H. Stone
Joseph L. Walsh
Hassler Whitney
David Widder

Known for

Ergodic theorem

George David Birkhoff (21 March 1884 — 12 November 1944) was an American mathematician, best known for
what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in
his generation, and during his prime he was considered by many to be the preeminent American mathematician.

The mathematician Garrett Birkhoff (1911—1996) was his son.

George Birkhoff 603

Career

Birkhoff obtained his A.B. and A.M. from Harvard. He completed his Ph.D. in 1907, on differential equations, at the
University of Chicago. While E. H. Moore was his supervisor, he was most influenced by the writings of Henri
Poincare. After teaching at the University of Wisconsin and Princeton University, he taught at Harvard University
from 1912 until his death.

Awards and honors

In 1923, he was awarded the inaugural Bocher Memorial Prize by the American Mathematical Society for his paper
Birkhoff (1917) containing, among other things, what is now called the Birkhoff curve shortening flow.

He was elected to the National Academy of Sciences, the American Philosophical Society, the American Academy
of Arts and Sciences, the Academie des Sciences in Paris, the Pontifical Academy, and the London and Edinburgh
Mathematical Societies.

Service

• Vice-president of the American Mathematical Society, 1919.

• President of the American Mathematical Society, 1925—1926.

• Editor of Transactions of the American Mathematical Society, 1920—1924.

Work

In 1912, attempting to solve the four color problem, Birkhoff introduced the chromatic polynomial. Even though this
line of attack did not prove fruitful, the polynomial itself became an important object of study in algebraic graph
theory.

In 1913, he proved Poincare's "Last Geometric Theorem," a special case of the three-body problem, a result that
made him world famous. In 1927, he published his Dynamical Systems . He wrote on the foundations of relativity
and quantum mechanics, publishing (with R E Langer) the monograph Relativity and Modern Physics in 1923. In
1923, Birkhoff also proved that the Schwarzschild geometry is the unique spherically symmetric solution of the
Einstein field equations. A consequence is that black holes are not merely a mathematical curiosity, but could result
from any spherical star having sufficient mass.

Birkhoff s most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining
insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a
fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics,
probability theory, group theory, and functional analysis. He also worked on number theory, the Riemann— Hilbert
problem, and the four colour problem. He proposed an axiomatization of Euclidean geometry different from Hilbert's
(see Birkhoff s axioms); this work culminated in his text Basic Geometry (1941).

In his later years, Birkhoff published two curious speculative works. His 1933 Aesthetic Measure proposed a
mathematical theory of aesthetics. While writing this book, he spent a year studying the art, music and poetry of
various cultures around the world. His 1938 Electricity as a Fluid combined his ideas on philosophy and science. His
1943 theory of gravitation is also puzzling, since Birkhoff knew (but didn't seem to mind) that his theory allows as
sources only matter which is a perfect fluid in which the speed of sound must equal the speed of light (which,
needless to say, is quite inconsistent with experiment!).

George Birkhoff 604

Influence on hiring practices

Albert Einstein and Norbert Wiener, among others, accused Birkhoff of advocating anti-Semitic hiring practices.
During the 1930s, when many Jewish mathematicians fled Europe and tried to obtain jobs in the USA, Birkhoff is
alleged to have influenced the hiring process at American institutions to exclude Jews. While Birkhoff may have
held anti-Semitic views, it was also the case that he had always been outspoken in his promotion of American
mathematics and mathematicians. It has been argued that Birkhoff s actions were in good part motivated by a desire
to assure jobs for home-grown American mathematicians. Saunders Mac Lane (1994), a close friend and collaborator
of Birkhoff s son, argued that any anti-Semitic tendencies Birkhoff may have had were not unusual for his time.

Selected publications

• Birkhoff, George David. 1913. "Proof of Poincare's geometric theorem" Trans. Amer. Math. Soc. 14: 14—22.

• Birkhoff, George David. 1917. "Dynamical Systems with Two Degrees of Freedom" Trans. Amer. Math. Soc. 18:
199-300.

• Birkhoff, George David and Ralph Beatley. 1959. Basic Geometry 3rd ed. Chelsea Publishing Co. [Reprint:
American Mathematical Society, 2000. ISBN 9780821821015]

References

• Aubin, David, 2005, "Dynamical systems" in Grattan-Guinness, I., ed., Landmark Writings in Western
Mathematics. Elsevier: 871—81.

• Saunders Mac Lane, 1994, "Jobs in the 1930s and the views of George D. Birkhoff," Math. Intelligencer 16:
9-10.

• Kip Thome, 19nn. Black Holes and Time Warps. W. W. Norton. ISBN 0-393-31276-3.

• Vandiver, H. S., 1963, "Some of my recollections of George David Birkhoff," /. Math. Anal. Appl. 7: 271—83.

• Norbert Wiener, 1956. / am a Mathematician. MIT Press. Especially pp. 27—28.

Further reading

• Morse, Marston (1970—80). "Birkhoff, George David". Dictionary of Scientific Biography. 2. New York: Charles
Scribner's Sons. pp. 143-146. ISBN 0684101149.

External links

T21

• O'Connor, John J.; Robertson, Edmund F., "George Birkhoff" , MacTutor History of Mathematics archive,

University of St Andrews.

T31

• George Birkhoff at the Mathematics Genealogy Project

• Birkhoff s biography - from National Academies Press, by Oswald Veblen.

References

[1] http://www.ams.org/online_bks/coll9/

[2] http://www-history.mcs.st-andrews.ac.uk/Biographies/Birkhoff.html

[3] http://www. genealogy. ams.org/id.php?id=5879

[4] http://darwin.nap.edu/books/0309082811/html/45.html

Stephen Weinberg

605

Stephen Weinberg

Steven Weinberg

Born

Steven Weinberg at the 2010 Texas Book Festival.

May 3, 1933New York City, New York, USA

Residence
Nationality

Fields
Institutions

Alma mater

Doctoral advisor

Doctoral students

Known for

Influenced
Notable awards

United States
United States

Theoretical Physics

University of California, Berkeley

MIT

Harvard University

University of Texas at Austin

Cornell University
Princeton University

Sam Treiman

Orlando Alvarez
Claude Bernard
Lay Nam Chang
Bob Holdom
Ubirajara van Kolck
Rafael Lopez-Mobilia
John Preskill
Fernando Quevedo
Mark G. Raizen
Scott Willenbrock

Electromagnetism and Weak Force

unification

Weinberg- Witten theorem

Alan Guth

Nobel Prize in Physics (1979)

Notes

He is married to the professor of law, Louise Weinberg.

Steven Weinberg (born May 3, 1933) is an American theoretical physicist and Nobel laureate in Physics for his
contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic
interaction between elementary particles.

Stephen Weinberg 606

Biography

Steven Weinberg was born in 1933 in New York City to Jewish immigrants Frederick and Eva Weinberg, but is an
atheist. He graduated from Bronx High School of Science in 1950 in the same graduating class as Sheldon
Glashow, whose own research, independent of Weinberg's, would result in them (and Abdus Salam) sharing the
same 1979 Nobel in Physics (see below).

Weinberg received his bachelor's degree from Cornell University in 1954, living at the Cornell branch of the
Telluride Association. He left Cornell and went to the Niels Bohr Institute in Copenhagen where he started his
graduate studies and research. After one year, Weinberg returned to Princeton University where he earned his Ph.D.
degree in Physics in 1957, studying under Sam Treiman.

Academic career

After completing his Ph.D., Weinberg worked as a post-doctoral researcher at Columbia University (1957—1959) and
University of California, Berkeley (1959) and then he was promoted to faculty at Berkeley (1960—1966). He did
research in a variety of topics of particle physics, such as the high energy behavior of quantum field theory,
symmetry breaking, pion scattering, infrared photons and quantum gravity. It was also during this time that he
developed the approach to quantum field theory that is described in the first chapters of his book The Quantum
Theory of Fields and started to write his textbook Gravitation and Cosmology . Both textbooks, perhaps especially
the second, are among the most influential texts in the scientific community in their subjects.

In 1966, Weinberg left Berkeley and accepted a lecturer position at Harvard. In 1967 he was a visiting professor at
MIT. It was in that year at MIT that Weinberg proposed his model of unification of electromagnetism and of nuclear
weak forces (such as those involved in beta-decay and kaon-decay), with the masses of the force-carriers of the
weak part of the interaction being explained by spontaneous symmetry breaking. One of its fundamental aspects was
the prediction of the existence of the Higgs boson. Weinberg's model, now known as the electroweak unification
theory, had the same symmetry structure as that proposed by Glashow in 1961: hence both models included the
then-unknown weak interaction mechanism between leptons, known as neutral current and mediated by the Z boson.
The 1973 experimental discovery of this Z boson was one verification of the electroweak unification. The paper by
Weinberg in which he presented this theory was one of the highest cited theoretical works ever in high energy
physics as of 2009.

After his 1967 seminal work on the unification of weak and electromagnetic interactions, Steven Weinberg
continued his work in many aspects of particle physics, quantum field theory, gravity, supersymmetry, superstrings
and cosmology, as well as a theory called Technicolor.

In the years after 1967, the full Standard Model of elementary particle theory was developed through the work of
many contributors. In it, the weak and electromagnetic interactions already unified by the work of Weinberg, Abdus
Salam and Sheldon Glashow, are made consistent with a theory of the strong interactions between quarks, in one
overarching theory. In 1973 Weinberg proposed a modification of the Standard Model which did not contain that
model's fundamental Higgs boson.

Weinberg became Higgins Professor of Physics at Harvard University in 1973.

It is of special importance that in 1979 he pioneered the modern view on the renormalization aspect of quantum field
theory that considers all quantum field theories as effective field theories and changed completely the viewpoint of
previous work (including his own in his 1967 paper) that a sensible quantum field theory must be renormalizable.
This approach allowed the development of effective theory of quantum gravity, low energy QCD, heavy quark
effective field theory and other developments, and it is a topic of considerable interest in current research.

In 1979, some six years after the experimental discovery of the neutral currents — i.e. the discovery of the inferred
existence of the Z boson — but following the 1978 experimental discovery of the theory's predicted amount of parity
violation due to Z bosons' mixing with electromagnetic interactions, Weinberg was awarded the Nobel Prize in

Stephen Weinberg 607

Physics, together with Sheldon Glashow, and Abdus Salam who had independently proposed a theory of electroweak
unification based on spontaneous symmetry breaking.

In 1982 Weinberg moved to the University of Texas at Austin as the Jack S. Josey-Welch Foundation Regents Chair
in Science and founded the Theory Group of the Physics Department.

ro]

There is current (2008) interest in Weinberg's 1976 proposal of the existence of new strong interactions — a
proposal dubbed "Technicolor" by Leonard Susskind — because of its chance of being observed in the LHC as an
explanation of the hierarchy problem.

Steven Weinberg's influence and importance are confirmed by the fact that he is frequently among the top scientists
with highest research effect indices, such as the h-index and the creativity index.

Other intellectual legacy

Besides his scientific research, Steven Weinberg has been a prominent public spokesman for science, testifying
before Congress in support of the Superconducting Super Collider, writing articles for the New York Review of
Books, and giving various lectures on the larger meaning of science. His books on science written for the public
combine the typical scientific popularization with what is traditionally considered history and philosophy of science
and atheism.

Weinberg was a major participant in what is known as the Science Wars, standing with Paul R. Gross, Norman
Levitt, Alan Sokal, Lewis Wolpert, and Richard Dawkins, on the side arguing for the hard realism of science and
scientific knowledge and against the constructionism proposed by such social scientists as Stanley Aronowitz, Barry
Barnes, David Bloor, David Edge, Harry Collins, Steve Fuller, and Bruno Latour.

Weinberg is also known for his support of Israel. He wrote an essay titled "Zionism and Its Cultural Adversaries" to
explain his views on the issue.

Weinberg has canceled trips to universities in the United Kingdom because of British boycotts directed towards
Israel. He has explained:

"Given the history of the attacks on Israel and the oppressiveness and aggressiveness of other countries in the
Middle East and elsewhere, boycotting Israel indicated a moral blindness for which it is hard to find any
explanation other than antisemitism.

His views on religion were expressed in a speech from 1999 in Washington, D.C.:

" With or without religion, good people can behave well and bad people can do evil; but for good people to do

ri2i

evil — that takes religion. "
He has also said:

ri3i

"The more the universe seems comprehensible, the more it seems pointless.
He attended and was a speaker at the Beyond Belief symposium in November 2006.

Personal

He is married to Louise Weinberg and has one daughter, Elizabeth.

Honors and awards

The honors and awards that Professor Weinberg received include:

• Honorary Doctor of Science degrees from a dozen institutions: University of Chicago, Knox College, University
of Rochester, Yale University, City University of New York, Dartmouth College, Weizmann Institute, Clark
University, Washington College, Columbia University, Bates College.

• American Academy of Arts and Sciences, elected 1968

Stephen Weinberg 608

National Academy of Sciences, elected 1972

J. R. Oppenheimer Prize, 1973

Dannie Heineman Prize for Mathematical Physics, 1977

Steel Foundation Science Writing Award, 1977, for authorship of The First Three Minutes (1977)

Elliott Cresson Medal (Franklin Institute), 1979

Nobel Prize in Physics, 1979

Elected to American Philosophical Society, Royal Society of London (Foreign Honorary Member), Philosophical

Society of Texas

James Madison Medal of Princeton University, 1991

National Medal of Science, 1991

Lewis Thomas Prize for Writing about Science, 1999.

2002 Humanist of the Year, American Humanist Association

James Joyce Award, University College Dublin, 2009

Selected publications

Bibliography: books authored / coauthored

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972)

The First Three Minutes: A Modern View of the Origin of the Universe (1977, updated with new afterword in

1993, ISBN 0-465-02437-8)

The Discovery of Subatomic Particles (1983)

Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures (1987; with Richard Feynman)

Dreams of a Final Theory: The Search for the Fundamental Laws of Nature (1993), ISBN 0-09-922391-0

The Quantum Theory of Fields (three volumes: 1995, 1996, 2003)

Facing Up: Science and Its Cultural Adversaries (2001, 2003, HUP)

Glory and Terror: The Coming Nuclear Danger (2004, NYRB)

Cosmology (2008, OUP)

Lake Views: This World and the Universe (2010), Belknap Press of Harvard University Press, ISBN 0674035151.

Scholarly articles

• Weinberg, S., A Model of Leptons [14] , Phys. Rev. Lett. 19, 1264-1266 (1967) - the electroweak unification
paper.

• Weinberg, S. & G. Feinberg. "Law of Conservation of Muons" , Columbia University, University of
California-Berkeley, United States Department of Energy (through predecessor agency the Atomic Energy
Commission), (Feb. 1961).

• Pais, A., Weinberg, S., Quigg, C, Riordan, M., Panofsky, W.K.H. & V. Trimble. "100 years of elementary
particles" , Stanford Linear Accelerator Center United States Department of Energy, Beam Line, vol. 27, issue

1, Spring 1997. (April 1, 1997).

ri7i

• Weinberg, S. "Pions in Large N Quantum Chromodynamics" , Phys. Rev. Lett. 105, 261601 (December 20,

2010)

Stephen Weinberg 609

Further reading

• Claude E. Shannon: A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27,
pp. 379-423, 623-656, 1948. (http://bstj.bell-labs.com/BSTJ/images/Vol27/bstj27-3-379.pdf) (http://bstj.
bell-labs.com/BSTJ/images/Vol27/bstj27-4-623.pdf)

• Claude E. Shannon and Warren Weaver: The Mathematical Theory of Communication. The University of Illinois
Press, Urbana, Illinois, 1949. ISBN 0-252-72548-4

• Rethnakaran Pulikkoonattu - Eric W. Weisstein: Mathworld biography of Shannon, Claude Elwood (1916—2001)
(http://scienceworld.wolfram.com/biography/Shannon.html)

• Claude E. Shannon: Programming a Computer for Playing Chess, Philosophical Magazine, Ser.7, Vol. 41, No.
314, March 1950. (Available online under External links below)

• David Levy: Computer Gamesmanship: Elements of Intelligent Game Design, Simon & Schuster, 1983. ISBN
0-671-49532-1

• Mindell, David A., "Automation's Finest Hour: Bell Labs and Automatic Control in World War II", IEEE Control
Systems, December 1995, pp. 72—80.

• David Mindell, Jerome Segal, Slava Gerovitch, "From Communications Engineering to Communications Science:
Cybernetics and Information Theory in the United States, France, and the Soviet Union" in Walker, Mark (Ed.),
Science and Ideology: A Comparative History, Routledge, London, 2003, pp. 66—95.

• Poundstone, William, Fortune's Formula, Hill & Wang, 2005, ISBN 978-0-8090-4599-0

• Gleick, James, The Information: A History, A Theory, A Flood, Pantheon, 201 1, ISBN 9780375423727

Shannon videos

• Shannon's video machines (http://www.youtube.com/watch?v=sBHGzRxfeJY)

• Shannon - father of the information age (http://www.youtube.com/watch?v=z2Whj_nL-x8)

• AT&T Tech Channel's Tech Icons - Claude Shannon (http://techchannel.att.com/play-video.cfm/201 1/4/19/
Tech-Icons-Claude-Shannon)

External links

• C. E. Shannon, An algebra for theoretical genetics, Massachusetts Institute of Technology, Ph.D. Thesis,
MIT-THESES// 1940-3 (1940) Online text at MIT (http://hdl.handle.net/1721. 1/11174)

• Shannon's math genealogy (http://www.genealogy.math.ndsu. nodak.edu/id. php?id=42920)

Claude Shannon 618

Shannon's NNDB profile (http://www.nndb.com/people/934/000023865/)

Works by or about Claude Shannon (http://worldcat.org/identities/lccn-n92-78142) in libraries (WorldCat

catalog)

A Mathematical Theory of Communication (http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html)

Communication Theory of Secrecy Systems (http://netlab.cs.ucla.edu/wiki/files/shannonl949.pdf)

Communication in the Presence of Noise (http://www.stanford.edu/class/eel04/shannonpaper.pdf)

Summary of Shannon's life and career (http://www.lucent.com/minds/infotheory/who.html)

Biographical summary from Shannon's collected papers (http://www.research.att.com/~njas/doc/shannonbio.

html)

Video documentary: "Claude Shannon - Father of the Information Age" (http://www.ucsd.tv/search-details.

asp?showID=6090)

Mathematical Theory of Claude Shannon (http://web.mit. edu/6.933/www/Fall2001/Shannonl.pdf) In-depth

MIT class paper on the development of Shannon's work to 1948.

Retrospective at the University of Michigan (http://www.engin.umich.edu/150th/alum-legends/shannon.

html)

Shannon's University of Michigan profile (http://www.engin.umich.edu/alumni/engineer/04SS/

achievements/advances. html#shannon)

Notes on Computer-Generated Text (http://www.nightgarden.com/infosci.htm)

Shannon's Juggling Theorem and Juggling Robots (http://www2.bc.edu/~lewbel/Shannon.html)

Color Photo of Shannon, Juggling (http://www.stanstudio.com/Boston_Photographers/portfolio/nw_8.htm)

Shannon's paper on computer chess, text (http://www.pi.infn.it/~carosi/chess/shannon.txt)

Shannon's paper on computer chess (http://www.ascotti.org/programming/chess/Shannon - Programming a

computer for playing chess. pdf)PDF (175 KiB)

Shannon's paper on computer chess, text, alternate source (http://www.dcc.uchile.cl/~cgutierr/cursos/IA/

shannon.txt)

A Bibliography of His Collected Papers (http://www.research.att.com/~njas/doc/shannonbib.html)

A Register of His Papers in the Library of Congress (http://memory.loc.gov/cgi-bin/query/r7faid/

faid: @ field(DOCID+ms00307 1 ))

The Technium: The (Unspeakable) Ultimate Machine (http://www.kk.org/thetechnium/archives/2008/03/

the_unspeakable.php)

The Most Beautiful Machine, (http://www.kugelbahn.ch/sesam_e.htm) (aka the "Ultimate Machine") It's a

communication based on the functions ON and OFF.

Guizzo, "The Essential Message: Claude Shannon and the Making of Information Theory" (http://dspace.mit.

edu/bitstream/1721.1/39429/l/54526133.pdf)

Claude Shannon, Edward O. Thorp, Fortune's Formula (http://www.fortunesformula.com)

Claude Shannon : Founding Father of Electronic Communication age,Dream 2047, December,2006, Shivaprasad

Khened (http://www.vigyanprasar.gov.in/dream/dec2006/Eng December.pdf)

Ludwig von Bertalanffy 619

Ludwig von Bertalanffy

Born 19 September 1901Vienna, Austria

Died 12 June 1972 (aged 70)Buffalo, New York, USA

Fields Biology and systems theory

Alma

University of Vienna
mater

Known for General System Theory

Influences Rudolf Carnap, Gustav Theodor Fechner, Nicolai Hartmann, Otto Neurath, Moritz Schlick

Influenced Russell L. Ackoff, Kenneth E. Boulding, Peter Checkland, C. West Churchman, Jay Wright Forrester, Ervin Laszlo, James Grier
Miller, Anatol Rapoport

Karl Ludwig von Bertalanffy (September 19, 1901, Atzgersdorf near Vienna, Austria — June 12, 1972, Buffalo,
New York, USA) was an Austrian-born biologist known as one of the founders of general systems theory (GST).
GST is an interdisciplinary practice that describes systems with interacting components, applicable to biology,
cybernetics, and other fields. Bertalanffy proposed that the laws of thermodynamics applied to closed systems, but
not necessarily to "open systems," such as living things. His mathematical model of an organism's growth over time,
published in 1934, is still in use today.

Von Bertalanffy grew up in Austria and subsequently worked in Vienna, London, Canada and the USA.

Biography

Ludwig von Bertalanffy was born and grew up in the little village of Atzgersdorf (now Liesing) near Vienna. The
Bertalanffy family had roots in the 16th century nobility of Hungary which included several scholars and court
officials. His grandfather Charles Joseph von Bertalanffy (1833—1912) had settled in Austria and was a state
theatre director in Klagenfurt, Graz, and Vienna, which were important positions in imperial Austria. Ludwig's father
Gustav von Bertalanffy (1861—1919) was a prominent railway administrator. On his mother's side Ludwig's
grandfather Joseph Vogel was an imperial counsellor and a wealthy Vienna publisher. Ludwig's mother Charlotte

Vogel was seventeen when she married the thirty-four year old Gustav. They divorced when Ludwig was ten, and

T21
both remarried outside the Catholic Church in civil ceremonies.

Ludwig von Bertalanffy grew up as an only child educated at home by private tutors until he was ten. When he went
to the gymnasium/grammar school he was already well trained in self study, and kept studying on his own. His
neighbour, the famous biologist Paul Kammerer, became a mentor and an example to the young Ludwig. In 1918
he started his studies at the university level with the philosophy and art history, first at the University of Innsbruck
and then at the University of Vienna. Ultimately, Bertalanffy had to make a choice between studying philosophy of
science and biology, and chose the latter because, according to him, one could always become a philosopher later,
but not a biologist. In 1926 he finished his PhD thesis (translated title: Fechner and the problem of integration of
higher order) on the physicist and philosopher Gustav Theodor Fechner.

Von Bertalanffy met his future wife Maria in April 1924 in the Austrian Alps, and were almost never apart for the
next forty-eight years. She wanted to finish studying but never did, instead devoting her life to Bertalanffy's career.
Later in Canada she would work both for him and with him in his career, and after his death she compiled two of
Bertalanffy's last works. They had one child, who would follow in his father's footsteps by making his profession in
the field of cancer research.

Ludwig von Bertalanffy 620

Von Bertalanffy was a professor at the University of Vienna from 1934—48, University of London (1948—49),
Universite de Montreal (1949), University of Ottawa (1950-54), University of Southern California (1955-58), the
Menninger Foundation (1958—60), University of Alberta (1961—68), and State University of New York at Buffalo
(SUNY) (1969-72). In 1972, he died from a sudden heart attack.

Work

Today, Bertalanffy is considered to be a founder and one of the principal authors of the interdisciplinary school of
thought known as general systems theory. According to Weckowicz (1989), he "occupies an important position in
the intellectual history of the twentieth century. His contributions went beyond biology, and extended into
cybernetics, education, history, philosophy, psychiatry, psychology and sociology. Some of his admirers even
believe that this theory will one day provide a conceptual framework for all these disciplines". Spending most of
his life in semi-obscurity, Ludwig von Bertalanffy may well be the least known intellectual titan of the twentieth

[5]

century.

The individual growth model

The individual growth model published by von Bertalanffy in 1934 is widely used in biological models and exists in
a number of permutations.

In its simplest version the so-called von Bertalanffy growth equation is expressed as a differential equation of length
(L) over time (t):

L'(t) = r B {L x - L(t))

when Tg is the von Bertalanffy growth rate and L^ the ultimate length of the individual. This model was proposed

earlier by A. Putter in 1920 (Arch. Gesamte Physiol. Mensch. Tiere, 180: 298-340).

The Dynamic Energy Budget theory provides a mechanistic explanation of this model in the case of isomorphs that

experience a constant food availability. The inverse of the von Bertalanffy growth rate appears to depend linearly on

the ultimate length, when different food levels are compared. The intercept relates to the maintenance costs, the

slope to the rate at which reserve is mobilized for use by metabolism. The ultimate length equals the maximum

length at high food availabilities.

Ludwig von Bertalanffy

621

Bertalanffy Module

To honor Bertalanffy, ecological systems engineer and scientist
Howard T. Odum named the storage symbol of his General Systems
Language as the Bertalanffy module (see image right).

General System Theory (GST)

The biologist is widely recognized for his contributions to science as a
systems theorist; specifically, for the development of a theory known
as General System Theory (GST). The theory attempted to provide
alternatives to conventional models of organization. GST defined new
foundations and developments as a generalized theory of systems with
applications to numerous areas of study, emphasizing holism over
reductionism, organism over mechanism.

Open systems

Energy Storage System

— ■ — *(0)

"EoLjrce" Qo"

"Store"

Passive electrical equivalent

V^vW—

QO---,, Ml

r*- 1 1 V i I

citor

Ar:flpl*l kirn H 1 O.lurn (1»1) Fig. 3-B, p. ':

Passive electrical schematic of the Bertalanffy

module together with equivalent expression in the

Energy Systems Language

Bertalanffy's contribution to systems theory is best known for his

theory of open systems. The system theorist argued that traditional

closed system models based on classical science and the second law of thermodynamics were untenable. Bertalanffy

maintained that "the conventional formulation of physics are, in principle, inapplicable to the living organism being

open system having steady state. We may well suspect that many characteristics of living systems which are

roi

paradoxical in view of the laws of physics are a consequence of this fact." However, while closed physical
systems were questioned, questions equally remained over whether or not open physical systems could justifiably
lead to a definitive science for the application of an open systems view to a general theory of systems.

In Bertalanffy's model, the theorist defined general principles of open systems and the limitations of conventional
models. He ascribed applications to biology, information theory and cybernetics. Concerning biology, examples
from the open systems view suggested they "may suffice to indicate briefly the large fields of application" that could
be the outlines of a wider generalization; from which, a hypothesis for cybernetics. Although potential

applications exist in other areas, the theorist developed only the implications for biology and cybernetics. Bertalanffy
also noted unsolved problems, which included continued questions over thermodynamics, thus the unsubstantiated
claim that there are physical laws to support generalizations (particularly for information theory), and the need for
further research into the problems and potential with the applications of the open system view from physics.

Systems in the social sciences

In the social sciences, Bertalanffy did believe that general systems concepts were applicable, e.g. theories that had
been introduced into the field of sociology from a modern systems approach that included "the concept of general
system, of feedback, information, communication, etc." The theorist critiqued classical "atomistic" conceptions of
social systems and ideation "such as 'social physics' as was often attempted in a reductionist spirit." Bertalanffy
also recognized difficulties with the application of a new general theory to social science due to the complexity of
the intersections between natural sciences and human social systems. However, the theory still encouraged for new
developments from sociology, to anthropology, economics, political science, and psychology among other areas.
Today, Bertalanffy's GST remains a bridge for interdisciplinary study of systems in the social sciences.

Ludwig von Bertalanffy 622

Publications
By Bertalanffy

• 1928, Kritische Theorie der Formbildung, Borntraeger. In English: Modern Theories of Development: An
Introduction to Theoretical Biology, Oxford University Press, New York: Harper, 1933

• 1928, Nikolaus von Kues, G. Mtiller, Munchen 1928.

• 1930, Lebenswissenschaft und Bildung, Stenger, Erfurt 1930

• 1937, Das Gefiige des Lebens, Leipzig: Teubner.

• 1940, Vom Molekiil zur Organismenwelt, Potsdam: Akademische Verlagsgesellschaft Athenaion.

• 1949, Das biologische Weltbild, Bern: Europaische Rundschau. In English: Problems of Life: An Evaluation of
Modern Biological and Scientific Thought, New York: Harper, 1952.

• 1953, Biophysik des Fliessgleichgewichts, Braunschweig: Vieweg. 2nd rev. ed. by W. Beier and R. Laue, East
Berlin: Akademischer Verlag, 1977

• 1953, "Die Evolution der Organismen", in Schopfungsglaube und Evolutionstheorie, Stuttgart: Alfred Kroner
Verlag, pp 53-66

• 1955, 'An Essay on the Relativity of Categories." Philosophy of Science, Vol. 22, No. 4, pp. 243—263.

• 1959, Stammesgeschichte, Umwelt und Menschenbild, Schriften zur wissenschaftlichen Weltorientierung Vol 5.
Berlin: Liittke

• 1962, Modern Theories of Development, New York: Harper

• 1967, Robots, Men and Minds: Psychology in the Modern World, New York: George Braziller, 1969 hardcover:
ISBN 0-8076-0428-3, paperback: ISBN 0-8076-0530-1

• 1968, General System theory: Foundations, Development, Applications, New York: George Braziller, revised
edition 1976: ISBN 0-8076-0453-4

• 1968, The Organismic Psychology and Systems Theory, Heinz Werner lectures, Worcester: Clark University
Press.

• 1975, Perspectives on General Systems Theory. Scientific-Philosophical Studies, E. Taschdjian (eds.), New York:
George Braziller, ISBN 0-8076-0797-5

• 1981, A Systems View of Man: Collected Essays, editor Paul A. LaViolette, Boulder: Westview Press, ISBN
0-86531-094-7

The first articles from Bertalanffy on General Systems Theory:

• 1945, Zu einer allgemeinen Systemlehre, Blatter fur deutsche Philosophic, 3/4. (Extract in: Biologia Generalis, 19
(1949), 139-164.

• 1950, An Outline of General System Theory, British Journal for the Philosophy of Science 1, p. 139-164

• 1951, General system theory - A new approach to unity of science (Symposium), Human Biology, Dec 1951, Vol.
23, p. 303-361.

About Bertalanffy

ri2i

• Sabine Brauckmann (1999). Ludwig von Bertalanffy (1901--1972) , ISSS Luminaries of the Systemics
Movement, January 1999.

• Peter Corning (2001). Fulfilling von Bertalanffy 's Vision: The Synergism Hypothesis as a General Theory of
Biological and Social Systems [13] , ISCS 2001.

• Mark Davidson (1983). Uncommon Sense: The Life and Thought of Ludwig Von Bertalanffy, Los Angeles: J. P.

Tarcher.

ri4i

• Debora Hammond (2005). Philosophical and Ethical Foundations of Systems Thinking , tripleC 3(2):

pp. 20-27. (Dead Link)

Ludwig von Bertalanffy 623

• Ervin Laszlo eds. (1972). The Relevance of General Systems Theory: Papers Presented to Ludwig Von
Bertalanffy on His Seventieth Birthday, New York: George Braziller, 1972.

• David Pouvreau (2006). Une biographie non officielle de Ludwig von Bertalanffy (1901-1972) , Vienna

• David Pouvreau & Manfred Drack (2007). On the history of Ludwig von Bertalanffy 's "General Systemology",
and on its relationship to cybernetics, in: International Journal of General Systems, Volume 36, Issue 3 June
2007, pages 281 -337.

• Thaddus E. Weckowicz (1989). Ludwig von Bertalanffy (1901-1972): A Pioneer of General Systems Theory
Center for Systems Research Working Paper No. 89-2. Edmonton AB: University of Alberta, February 1989.

References

[I] T.E. Weckowicz (1989). Ludwig von Bertalanffy (1901-1972): A Pioneer of General Systems Theory (http://www.richardjung.cz/bertl.
pd). Working paper Feb 1989. p. 2

[2] Mark Davidson (1983). Uncommon Sense: The Life and Thought of Ludwig Von Bertalanffy. Los Angeles: J. P. Tarcher. p.49

[3] Bertalanffy Center for the Study of Systems Science, page: His Life - Bertalanffy's Origins and his First Education (http://www. bertalanffy.

org/c_71.html). Retrieved 2009-04-27
[4] Davidson p. 51
[5] Davidson, p. 9.
[6] Bertalanffy, L. von, (1934). Untersuchungen tiber die Gesetzlichkeit des Wachstums. I. Allgemeine Grundlagen der Theorie; mathematische

und physiologische Gesetzlichkeiten des Wachstums bei Wassertieren. Arch. Entwicklungsmech., 131:613-652.
[7] Nicholas D. Rizzo William Gray (Editor), Nicholas D. Rizzo (Editor), (1973) Unity Through Diversity. A Festschrift for Ludwig von

Bertalanffy. Gordon & Breach Science Pub
[8] Bertalanffy, L. von, (1969). General System Theory. New York: George Braziller, pp. 39-40
[9] Bertalanffy, L. von, (1969). General System Theory. New York: George Braziller, pp. 139-1540
[10] Bertalanffy, L. von, (1969). General System Theory. New York: George Braziller, pp. 196

[II] Bertalanffy, L. von, (1969). General System Theory. New York: George Braziller, pp. 194-197
[12] http://isss.org/projects/ludwig_von_bertalanffy

[13] http://www.complexsystems.org/abstracts/vonbert.html

[14] http://triplec.uti. at/files/tripleC3(2)_Hammond.pdf

[15] http://www.bertalanffy.org

[16] http://www.richardjung.cz/bertl.pdf

External links

• International Society for the Systems Sciences' (http://www.isss.org/lumLVB.htm) biography of Ludwig von
Bertalanffy

• Bertalanffy Center for the Study of Systems Science (http://www.bertalanffy.org/) BCSSS in Vienna.

• Ludwig von Bertalanffy (1901-1972): A Pioneer of General Systems Theory (http://www.richardjung.cz/bertl.
pdf) working paper by T.E. Weckowicz, University of Alberta Center for Systems Research.

Stephen Smale

624

Stephen Smale

Born

July 15, 1930

Nationality

B^ United States

Fields

Mathematics

Institutions

City University of Hong Kong
University of Chicago
Columbia University
University of California, Berkeley

Alma mater

University of Michigan

Doctoral advisor

Raoul Bott

Doctoral students

Rufus Bowen
John Guckenheimer
Jacob Palis
Themistocles M. Rassias

Notable awards

Wolf Prize (2006/07)

National Medal of Science (1996)

Fields Medal (1966)

Steven Smale a.k.a. Steve Smale, Stephen Smale (born July 15, 1930) is an American mathematician from Flint,
Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty
of the University of California, Berkeley (1960-61 and 1964-1995).

Education and career

He entered the University of Michigan in 1948. Initially, Smale was a good student, placing into an honors calculus
sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with
mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as
a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly his
first years, earning a C average as a graduate student. It was only when the department chair, Hildebrant, threatened
to kick out Smale, that he began to work hard. Smale finally earned his Ph.D. in 1957, under Raoul Bott.

Smale began his career as an instructor at the college at the University of Chicago. In 1958, he astounded the
mathematical world with a proof of a sphere eversion. He then cemented his reputation with a proof of the Poincare
conjecture for all dimensions greater than or equal to 5; he later generalized the ideas in a 107 page paper that
established the h-cobordism theorem.

After having made great strides in topology, he then turned to the study of dynamical systems, where he made
significant advances as well. His first contribution is the Smale horseshoe that jumpstarted significant research in
dynamical systems. He also outlined a research program carried out by many others. Smale is also known for
injecting Morse theory into mathematical economics, as well as recent explorations of various theories of
computation.

In 1998 he compiled a list of 18 problems in mathematics to be solved in the 21st century, known as Smale's
problems. This list was compiled in the spirit of Hilbert's famous list of problems produced in 1900. In fact, Smale's
list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of
Hilbert's sixteenth problem, both of which are still unsolved. Other famous problems on his list include the Poincare
conjecture, the P = NP problem, and the Navier-Stokes equations, all of which have been designated Millennium

Stephen Smale 625

Prize Problems by the Clay Mathematics Institute.

Earlier in his career, Smale was involved in controversy over remarks he made regarding his work habits while
proving the higher dimensional Poincare conjecture. He said that his best work had been done "on the beaches of
Rio". This led to the withholding of his grant money from the NSF. He has been politically active in various
movements in the past, such as the Free Speech movement. At one time he was subpoenaed by the House
Un-American Activities Committee.

In 1960 Smale was appointed an associate professor of mathematics at the University of California, Berkeley,
moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at UC
Berkeley where he has spent the main part of his career. He retired from UC Berkeley in 1995 and took up a post as
professor at the City University of Hong Kong. He also amassed over the years one of the finest private mineral
collections in existence. Many of Smale's mineral specimens can be seen in the book - The Smale Collection: Beauty
in Natural Crystals. [1]

Since 2002 Smale is a Professor at the Toyota Technological Institute at Chicago; starting August 1, 2009, he is also

T21
a Distinguished University Professor at the City University of Hong Kong.

T31
In 2007, Smale was awarded the Wolf Prize in mathematics.

Important publications

• S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Annals of Mathematics, 2nd Ser.,
74 (1961), no. 2, 391 - 406. (via JSTOR [4] )

• S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747 —
817. ([5])

• F. Cucker & R Wong, The Collected Papers of Stephen Smale, ISBN 978-98 1-02-4307-4

• L. Blum, F. Cucker, M. Shub and S. Smale, Complexity and Real Computation, ISBN 0-387-98281-7.

References

[1] http://www.lithographie.org/bookshop/the_smale_collection.htm

[2] Stephen Smale Vita, (http://ttic.uchicago.edu/~smale/vita.html) Accessed November 18, 2009.

[3] Press release (http://www.huji.ac.il/cgi-bin/dovrut/dovrut_search_eng_dev.pl7mesgel 16895485932688760)

[4] http://links.jstor.org/sici?sici=0003-486X%28196109%292%3A74%3A2%3C391%3AGPCIDG%3E2.0.CO%3B2-B

[5] http://www.ams.org/bull/1967-73-06/S0002-9904-1967- 1 1797-X/home.html

External links

• Stephen Smale (http://www. genealogy. ams.org/id.php?id=5086) at the Mathematics Genealogy Project

• O'Connor, John J.; Robertson, Edmund F., "Stephen Smale" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Smale.html), MacTutor History of Mathematics archive, University of St Andrews.

• Weisstein, Eric W., " Smale's Problems (http://mathworld.wolfram.com/SmalesProblems.html)" from
MathWorld.

• Robion Kirby, Stephen Smale: The Mathematician Who Broke the Dimension Barrier (http://www.ams.org/
notices/20001 1/rev -kirby.pdf), a book review of a biography in the Notices of the AMS.

Stephen Smale 626

Personal Website at Universities

• Steven Smale (http://www.ee.cityu.edu.hk/~cccn/smale.htm) at the City University of Hong Kong

• Stephen Smale (http://www.cityu.edu.hk/ma/staff/smale.html) at the City University of Hong Kong

• Stephen Smale (http://ttic.uchicago.edu/~smale/vita.html) at the University of Chicago

• Steve Smale (http://math.berkeley.edu/~smale/) at the University of California, Berkeley

Yakov Sinai

627

Yakov Sinai

Yakov G. Sinai

Born

Yakov G. Sinai
September 21, 1935Moscow, Russian Soviet Federative Socialist Republic, USSR

Residence Princeton, New Jersey, United States

Nationality Russian / American

Fields Mathematician

Institutions Moscow State University, Princeton University

Alma mater Moscow State University

Doctoral advisor Andrey Kolmogorov

Doctoral students Leonid Bunimovich
Grigory Margulis
Marina Ratner

Known for

works on dynamical systems, mathematical and statistical physics, probability theory, mathematical fluid dynamics

Notable awards Boltzmann Medal (1986)

Dannie Heineman Prize (1990)
Dirac Prize (1992)
Wolf Prize (1997)
Nemmers Prize (2002)
Henri Poincare Prize (2009)

Yakov Grigorevich Sinai (Russian: 5Ikob TpHropbeBMH Oman; born September 21, 1935) is an influential
mathematician working in the theory of dynamical systems, in mathematical physics and in probability theory. His
work has shaped the modern metric theory of dynamical systems (also often called after Kolmogorov the theory of
stochasticity of dynamical systems). Sinai has created bridges connecting the world of deterministic (dynamical)
systems with the world of probabilistic (stochastic) systems.

Yakov Sinai 628

Biography

Sinai was born in Moscow, USSR (now Russia) into a Jewish family that played a prominent role in Russia's
scientific and cultural life since the nineteenth century. His grandfather Veniamin Kagan was a Russian geometer,
and Sinai's parents were prominent researchers in the medical and biological sciences.

Yakov Sinai received his Ph.D. from Moscow State University in 1960; his advisor was Andrey Kolmogorov. In
1971 he became a Professor at Moscow State University and a senior researcher at the Landau Institute of
Theoretical Physics. Since 1993 he has been a Professor of Mathematics at Princeton University.

Sinai is a member of the United States National Academy of Sciences, Russian Academy of Sciences and others.
Among his awards are the Boltzmann Medal (1986), Dannie Heineman Prize for Mathematical Physics (1990), Dirac
Medal (1992), the Wolf Prize in Mathematics (1997), Nemmers Prize (2002), and the Henri Poincare Prize (2009).
Sinai has worked, among other topics, on Kolmogorov— Sinai entropy, Sinai Billiards, Sinai's random walk,
Sinai— Ruelle— Bowen measures, Pirogov— Sinai theory.

Sinai is highly respected in the physics community, where, as well as in mathematics, Kolmogorov-Sinai entropy,
Sinai's billiards, Sinai's random walk, Sinai-Ruelle-Bowen measures, Pirogov-Sinai theory and his other
achievements are basic notions that shaped the understanding of many fundamental physical phenomena.

References

• Sinai biography

• Another biography

• Sinai on scholarpedia

• Yakov Sinai at the Mathematics Genealogy Project

References

[1] http://www. northwestern. edu/provost/awards/nemmers/nemprmath.html#sinai

[2] http://www.worldscibooks.com/contact/ysinai.shtml

[3] http://www.scholarpedia.Org/article/User:Sinai

[4] http://www. genealogy. ams.org/id.php?id=10481

Marston Morse

629

Marston Morse

H. C. Marston Morse

Born

Marston Morse in 1965 (courtesy MFO)
24 March 1892Waterville, Maine

Died

22 June 1977 (aged 85)Princeton,

New Jersey

Nationality

BB American

Fields

Mathematics

Institutions

Harvard University

Alma mater

Colby College
Harvard University

Doctoral advisor

G. D. Birkhoff

Doctoral students

Emilio Baiada
Gustav Hedlund
Walter Leighton
Sumner Myers

Known for

Morse theory

Harold Calvin Marston Morse (24 March 1892 — 22 June 1977) was an American mathematician best known for
his work on the calculus of variations in the large, a subject where he introduced the technique of differential
topology now known as Morse theory. In 1933 he was awarded the Bocher Memorial Prize for his work in
mathematical analysis.

He was born in Waterville, Maine to Ella Phoebe Marston and Howard Calvin Morse in 1892. He received his
bachelor's degree from Colby College (also in Waterville) in 1914. At Harvard University, he received both his
master's degree in 1915 and his Ph.D. in 1917.

He taught at Harvard, Brown, and Cornell Universities before accepting a position in 1935 at the Institute for
Advanced Study in Princeton, where he remained until his retirement in 1962.

He spent most of his career on a single subject, eponymously titled Morse Theory, a branch of differential topology.
Morse Theory is a very important subject in modern mathematical physics, such as string theory.

Marston Morse 630

Quotes

"Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition
of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's
sake and pulls it down to earth."

Publications

• Morse, Marston (1981), Bott, Raoul, ed., Selected papers , Berlin, New York: Springer- Verlag,

121
Morse, Marston (1987), Montgomery, Deane; Bott, Raoul, eds., Collected papers. Vol. 1--6 , Singapore: World

ISBN 978-0-387-90532-7, MR635124

• Morse, Marston (1987), Montgomery,
Scientific Publishing Co., ISBN 9789971978945, MR889255

External links

131

• O'Connor, John J.; Robertson, Edmund F., "Marston Morse" , MacTutor History of Mathematics archive,

University of St Andrews.
mi

• Marston Morse at the Mathematics Genealogy Project

References

[1] http://books.google.com/books?id=UOLuAAAAMAAJ

[2] http://books.google.com/books?id=6Bc8bwAACAAJ

[3] http://www-history.mcs.st-andrews.ac.uk/Biographies/Morse.html

[4] http://www. genealogy. ams.org/id.php?id=4926

G. A. Hedlund

Gustav Arnold Hedlund, an American mathematician, was one of the founders of symbolic and topological
dynamics. He was a student of Marston Morse.

William Ross Ashby

631

William Ross Ashby

W. Ross Ashby

Born 6 September 1903London, England

Died 15 November 1972 (aged 69)

Fields Psychiatry, Cybernetics, Systems theory

Known for Cybernetics, Law of Requisite Variety, Principle of Self-Organization

Influenced Norbert Wiener, Ludwig von Bertalanffy, Herbert Simon, Stafford Beer and Stuart Kauffman

W. Ross Ashby (London, 6 September 1903 — 15 November 1972) was an English psychiatrist and a pioneer in
cybernetics, the study of complex systems. His first name was not used: he was known as Ross Ashby.

His two books, Design for a brain and An introduction to cybernetics, were landmark works. They introduced exact,
logical, thinking to the nascent discipline, and were highly influential.

Biography

William Ross Ashby was born in 1903 in London, where his father was working at an advertising agency. From
1917 to 1921 William studied at the Edinburgh Academy in Scotland, and from 1921 at Sidney Sussex College,
Cambridge, where he received his B.A. in 1924 and his M.B. and B.Ch. in 1928. From 1924 to 1928 he worked at
the St. Bartholomew's Hospital in London. Later on he also received a Diploma in Psychological Medicine in 1931,
and an M.A. 1930 and M.D. from Cambridge in 1935.

Ross Ashby started working in 1930 as a Clinical Psychiatrist in the London County Council. From 1936 until 1947
he was a Research Pathologist in the St Andrew's Hospital in Northampton in England. From 1945 to 1947 he served
in India where he was a Major in the Royal Army Medical Corps.

When he returned to England he served as Director of Research of the Barnwood House Hospital in Gloucester from
1947 until 1959. For a year he was Director of the Burden Neurological Institute in Bristol. In 1960 he went to the
United States and became Professor, Depts. of Biophysics and Electrical Engineering, University of Illinois at
Urbana-Champaign, until his retirement in 1970.

Ashby was president of the Society for General Systems Research from 1962 to 1964. He became a fellow of the
Royal College of Psychiatry in 1971.

On March 4—6, 2004, a W. Ross Ashby centenary conference was held at the University of Illinois at
Urbana-Champaign to mark the 100th anniversary of his birth. Presenters at the conference included Stuart
Kauffman, Stephen Wolfram and George Klir. In February 2009 a special issue of the International Journal of
General Systems was specifically devoted to Ashby and his work, containing papers from leading scholars such as
Klaus Krippendorff, Stuart Umpleby and Kevin Warwick

[41

William Ross Ashby 632

Work

Despite being widely influential within cybernetics, systems theory and, more recently, complex systems, he is not
as well known as many of the notable scientists his work influenced including Herbert Simon, Norbert Wiener,
Ludwig von Bertalanffy, Stafford Beer and Stuart Kauffman.

Journal

Ashby kept a journal for over 44 years in which he recorded his ideas about new theories. He started May 1928,
when he was medical student at St. Bartholomew's Hospital in London. Over the years he wrote down a series of 25
volumes with intotal 7,400 pages. In 2003 these journals were given to The British Library, London, and since 2008,
they were made available online as The W. Ross Ashby Digital Archive.

Cybernetics

Ross Ashby was one of the original members of the Ratio Club, a small informal dining club of young psychologists,
physiologists, mathematicians and engineers who met to discuss issues in cybernetics. The club was founded in 1949
by the neurologist John Bates and continued to meet until 1958.

Earlier, in 1946, Alan Turing wrote a letter to Ashby suggesting he use Turing's Automatic Computing Engine

ro]

(ACE) for his experiments instead of building a special machine. In 1948 Ashby made the Homeostat.

Variety

In An Introduction to Cybernetics Ashby formulated his Law of Requisite Variety stating that "variety absorbs
variety, defines the minimum number of states necessary for a controller to control a system of a given number of
states." This law can be applied for example to the number of bits necessary in a digital computer to produce a
required description or model.

In response Conant (1970) produced his so called "Good Regulator theorem" stating that "every Good Regulator of a
System Must be a Model of that System". [10]

Stafford Beer applied Variety to found management cybernetics and the Viable System Model. Working
independently Gregory Chaitin followed this with algorithmic information theory.

Publications

Books

• 1952. Design for a Brain , Chapman & Hall.

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• 1956. An Introduction to Cybernetics , Chapman & Hall.

• 1981. Conant, Roger C. (ed.). Mechanisms of Intelligence: Ross Ashby' s Writings on Cybernetics, Intersystems
Publishers.

Articles, a selection

• 1940. "Adaptiveness and equilibrium". In: /. Ment. Sci. 86, 478.

• 1945. "Effects of control on stability". In: Nature, London, 155, 242-243.

• 1946. "The behavioural properties of systems in equilibrium". In: Amer. J. Psychol. 59, 682-686.

• 1947. "Principles of the Self-Organizing Dynamic System". In: Journal of General Psychology (1947). volume
37, pages 125-128.

• 1948. "The homeostat". In: Electron, 20, 380.

• 1962. "Principles of the Self-Organizing System". In: Heinz Von Foerster and George W. Zopf, Jr. (eds.),
Principles of Self-Organization (Sponsored by Information Systems Branch, U.S. Office of Naval Research).

ri3i

Republished as a PDF in Emergence: Complexity and Organization (E:CO) Special Double Issue Vol. 6, Nos.

William Ross Ashby 633

1-2 2004, pp. 102-126.
About W. Ross Ashby

• Asaro, Peter (2008). "From Mechanisms of Adaptation to Intelligence Amplifiers: The Philosophy ofW. Ross

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Ashby, ' in Michael Wheeler, Philip Husbands and Owen Holland (eds.) The Mechanical Mind in History,

Cambridge, MA: MIT Press.

References

[I] Biography of W. Ross Ashby (http://www.rossashby.info/biography.html) The W. Ross Ashby Digital Archive, 2008.

[2] Autobiographical summary (http://www.rossashby.info/autobiography.html), taken from Ashby's own notes, made about 1972.

[3] W. Ross Ashby Centenary Conference (http://www.rossashby.info/centenary.html) The W. Ross Ashby Digital Archive, 2008

[4] International Journal of General Systems (http://www.informaworld.com/smpp/title~content=t713642931~db=all)

[5] Cosma Shalizi, W. Ross Ashby (http://bactra.org/notebooks/ashby.html) web page, 1999.

[6] W. Ross Ashby Journal (1928-1972) (http://www.rossashby.info/journal/index.html) The W. Ross Ashby Digital Archive, 2008.

[7] Alan Turing letter (http://www.rossashby.info/letters/turing.html) The W. Ross Ashby Digital Archive, 2008.

[8] Java applet simulation (http://www.hrat.btinternet.co.uk/Homeostat.html) by Dr Horace Townsend

[9] (Ashby 1956)

[10] Int. J. Systems Sci., 1970. vol 1, No. 2 pp89-97

[II] http://www.archive.org/details/designforbrainorOOashb
[12] http://pespmcl.vub.ac.be/ASHBBOOK.html

[13] http://emergence.org/ECO_site/ECO_Archive/Issue_6_l-2/Ashby.pdf
[14] http://peterasaro.org/writing/Asaro%20Ashby.pdf

External links

• The W. Ross Ashby Digital Archive (http://www.rossashby.info/index.html) includes an extensive biography,
bibliography, letters, photographs, movies, and fully-indexed images of all 7,400 pages of Ashby's 25 volume
journal.

• Homepage of William Ross Ashby (http://www.gwu.edu/~asc/biographies/ashby/ashby.html) with a short
text from the Encyclopaedia Britannica Yearbook 1973, and some links.

• Asaro, Peter M. (2008). "From Mechanisms of Adaptation to Intelligence Amplifiers: The Philosophy of W. Ross
Ashby," (http://cybersophe.org/writing/Asaro Ashby.pdf) in Michael Wheeler, Philip Husbands and Owen
Holland (eds.) The Mechanical Mind in History (http://mitpress.mit.edu/catalog/item/default. asp ?ttype=2&
tid=11479), Cambridge, MA: MIT Press, pp. 149-184.

• W. Ross Ashby (http://bactra.org/notebooks/ashby.html) web page by Cosma Shalizi, 1999.

• W. Ross Ashby (1956): An Introduction to Cybernetics, (Chapman & Hall, London): available electronically
(http://pcp.lanl.gov/ASHBBOOK.html) , Principia Cybernetica Web, 1999

• The Law of Requisite Variety (http://pcp.lanl.gov/reqvar.html) in the Principia Cybernetica Web, 2001.

• 159 Aphorisms from Ashby and further links at the Cybernetics Society (http://www.cybsoc.org/ross.htm)

• W. Ross Ashby, Cybernetics and Requisite Variety (http://www.panarchy.org/ashby/variety. 1956.html)
(1956) from An Introduction to Cybernetics

• W. Ross Ashby, Feedback, Adaptation and Stability (http://www.panarchy.org/ashby/adaptation. 1960.html)
(1960) from Design for a Brain

• What is Cybernetics? (http://www.youtube.com/watch?v=_hjAXkNbPfk) Livas short introductory videos on
YouTube

Robert Rosen 634

Robert Rosen

Robert Rosen may refer to:

• Robert Rosen (born 1987), Swedish ice hockey player

• Robert Rosen (theoretical biologist) (1934 — 1998), American theoretical biologist

• Robert Rosen (writer) (born 1952), American author

• Robert Ozn (born Robert M. Rosen), American producer, screenwriter, and entertainer

Edward Norton Lorenz

635

Edward Norton Lorenz

Born

Died
Residence

Fields
Institutions

Edward Norton Lorenz

Alma mater

Doctoral advisor

Doctoral students
Known for

Notable awards

Edward Norton Lorenz
May 23, 1917West Hartford, Connecticut, United States

April 16, 2008 (aged 90)Cambridge, Massachusetts, United States
United States

Mathematics and Meteorology
Massachusetts Institute of Technology

Dartmouth College (BA, 1938)

Harvard University (Master's, 1940)

Massachusetts Institute of Technology (SM, 1943; ScD, 1948)

James Murdoch Austin

Kevin E. Trenberth

Chaos theory
Lorenz attractor
Butterfly effect

Kyoto Prize (1991)

Edward Norton Lorenz (May 23, 1917 - April 16, 2008) was an American mathematician and meteorologist, and a
pioneer of chaos theory. He discovered the strange attractor notion and coined the term butterfly effect.

Biography

[2]

Lorenz was born in West Hartford, Connecticut. He studied mathematics at both Dartmouth College in New
Hampshire and Harvard University in Cambridge, Massachusetts. From 1942 until 1946, he served as a weather
forecaster for the United States Army Air Corps. After his return from the war, he decided to study meteorology.
Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor
for many years. He was a Professor Emeritus at MIT from 1987 until his death

[l]

During the 1950s, Lorenz became skeptical of the appropriateness of the linear statistical models in meteorology, as
most atmospheric phenomena involved in weather forecasting are non-linear. His work on the topic culminated in
the publication of his 1963 paper Deterministic Nonperiodic Flow in Journal of the Atmospheric Sciences, and with
it, the foundation of Chaos theory. His description of the Butterfly effect followed in 1969, Kyoto Prize

for basic sciences, in the field of earth and planetary sciences, in 1991, the Buys Ballot Award in 2004, and the
Tomassoni Award in 2008. In his later years, he lived in Cambridge, Massachusetts. He was an avid outdoorsman,
who enjoyed hiking, climbing, and cross-country skiing. He kept up with these pursuits until very late in his life, and

Edward Norton Lorenz 636

managed to continue most of his regular activities until only a few weeks before his death. According to his
daughter, Cheryl Lorenz, Lorenz had "finished a paper a week ago with a colleague." On April 16, 2008, Lorenz

ro]

died at his home in Cambridge at the age of 90, having suffered from cancer.

Awards

1969 Carl Gustaf Rossby Research Medal, American Meteorological Society.
1973 Symons Memorial Gold Medal, Royal Meteorological Society.

1975 Fellow, National Academy of Sciences (U.S.A.).
1981 Member, Norwegian Academy of Science and Letters.

1983 Crafoord Prize, Royal Swedish Academy of Sciences.

1984 Honorary Member, Royal Meteorological Society.

1989 Elliott Cresson Medal, The Franklin Institute

1991 Kyoto Prize for "... his boldest scientific achievement in discovering "deterministic chaos" . '.
2004 Buys Ballot medal.

2004 Lomonosov Gold Medal

Work

Lorenz built a mathematical model of the way air moves around in the atmosphere. As Lorenz studied weather
patterns he began to realize that they did not always change as predicted. Minute variations in the initial values of
variables in his twelve variable computer weather model (c. 1960) would result in grossly divergent weather
patterns. This sensitive dependence on initial conditions came to be known as the butterfly effect (it also meant
that weather predictions from more than about a week out are generally fairly inaccurate).

Lorenz went on to explore the underlying mathematics and published his conclusions in a seminal work titled
Deterministic Nonperiodic Flow, in which he described a relatively simple system of equations that resulted in a very
complicated dynamical object now known as the Lorenz attractor.

Publications

Lorenz published several books and articles. A selection:

1955 Available potential energy and the maintenance of the general circulation. Tellus. Vol.7

1963 Deterministic nonperiodic flow. Journal of Atmospheric Sciences. Vol.20 : 130 — 141 link

1967 The nature and theory of the general circulation of atmosphere. World Meteorological Organization. No. 218

1969 Three approaches to atmospheric predictability. American Meteorological Society. Vol.50

1972 Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?

1976 Nondeterministic theories of climate change. Quaternary Research. Vol.6

1990 Can chaos and intransitivity lead to interannual variability? Tellus. Vol.42A

2005 Designing Chaotic Models. Journal of the Atmospheric Sciences: Vol. 62, No. 5, pp. 1574—1587.

Edward Norton Lorenz 637

References

[I] Tim Palmer (2008). "Edward Norton Lorenz". Physics Today 61 (9): 81-82. doi:10.1063/1.2982132.

[2] "Lorenz Receives 1991 Kyoto Prize" (http://web.mit.edu/newsoffice/tt/1991/24996/24998.html). MIT News Office. 1991. .
[3] Edward N. Lorenz (1963). "Deterministic Nonperiodic Flow" (http://journals.ametsoc.org/doi/pdf/10.1175/

1520-0469(1963)020<0130:DNF>2.0.CO;2). Journal of the Atmospheric Sciences 20: 130-141.

doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. .
[4] Edward N. Lorenz (1969). "Atmospheric predictability as revealed by naturally occurring analogues" (http://ams.allenpress.com/archive/

1520-0469/26/4/pdf/il520-0469-26-4-636.pdf). Journal of the Atmospheric Sciences 26: 636-646.

doi:10.1175/1520-0469(1969)26<636:APARBN>2.0.CO;2. .
[5] Edward N. Lorenz (1969). "Three approaches to atmospheric predictability" (http://eapsweb.mit.edu/research/Lorenz/

Three_approaches_1969.pdf). Bulletin of the American Meteorological Society 50: 345—349. .
[6] , Maggie Fox, Eric Walsh (2008). Edward Lorenz, father of chaos theory, dead at 90 (http://www.reuters.com/article/newsOne/

idUSN1632944820080416). Reuters. .
[7] Kenneth Chang (2008-04- 17). "Edward N. Lorenz, a Meteorologist and a Father of Chaos Theory, Dies at 90" (http://www.nytimes.com/

2008/04/ 17/us/171orenz.html?ref=us). New York Times. . Retrieved 2010-05-01.
[8] "Edward Lorenz, father of chaos theory, dies at age 90" (http://edition.cnn. com/2008/TECH/04/16/lorenz.obit.ap/?iref=hpmostpop).

CNN..
[9] The term was first recorded from Lorenz's address at the annual meeting of the American Association for the Advancement of Science, on

December 29, 1979.
[10] http://eapsweb.mit.edu/research/Lorenz/Deterministic_63.pdf

[II] According to the Web of Science online academic database, this paper has received at least 4000 unique citations by subsequent authors,
making it one of the most-cited papers of all time.

External links

• List of Lorenz's publications, including downloadable pdfs (MIT website) (http://eapsweb.mit.edu/research/
Lorenz/publications.htm)

• Video clip of Edward N. Lorenz speaking at the International Conference on Complex Systems, hosted by the
New England Complex Systems Institute (NECSI) (http://www.necsi.edu/events/iccs/video/
iccs2002wednesday/3-lorenzclip.html)

• Obituary (http://www.telegraph.co.uk/news/main.jhtml7xmWnews/2008/04/18/dbl801.xml), Daily
Telegraph, 18 April 2008.

Otto Rossler 638

Otto Rossler

Otto E. Rossler (born 20 May 1940) is a German biochemist and is notable for his work on chaos theory and his
theoretical equation known as the Rossler attractor.

Biography

Rossler was born in Berlin. He was awarded his MD in 1966. Rossler then began his post doc at the Max Planck
Institute for Behavioral Physiology, in Bavaria. In 1969, he started a visiting appointment at the Center for
Theoretical Biology at SUNY-Buffalo. Later that year, he became Professor for Theoretical Biochemistry at the
University of Tubingen. In 1976, he became a tenured University Docent. In 1994, he became Professor of
Chemistry by decree.

Rossler has held visiting positions at the University of Guelph (Mathematics) in Canada, the Center for Nonlinear
Studies of the University of California at Los Alamos, the University of Virginia (Chemical Engineering), the
Technical University of Denmark (Theoretical Physics), and the Santa Fe Institute (Complexity Research) in New
Mexico.

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In June 2008 Rossler emerged in the public eye as a critic of the Large Hadron Collider (LHC) proton collision

experiment supervised by the European Organization for Nuclear Research in Geneva and was involved in a failed

law suit to halt its start up.

Rossler has authored around 300 scientific papers in fields as wide-ranging as biogenesis, the origin of language,
differentiable automata, chaotic attractors, endophysics, micro relativity, artificial universes, the hypertext
encyclopedia, and world-changing technology.

Bibliography

Encounter with Chaos, 1992, (ISBN 0-38755-647-8)

Endophysics: The world As an Interface, 1992, (ISBN 9-81022-752-3)

Jonas World — The Thinking of Child, 1994,

The Flaming Sword, 1996, (ISBN 3-7165-1017-3)

with Rene Stettler: Interventionen. Vertikale und horizontale Grenziiberschreitung. 1997, (ISBN 3-87877-627-6)

with Peter Weibel: Aussenwelt — Innenwelt — Uberwelt. Ein Gesprdch. 1997, (ISBN 3-87877-628-4)

with Wilfried Kriese: Mut zu Lampsacus. Das Internet als Chance. 1998

with Artur P. Schmidt: Medium des Wissens. Das Menschenrecht auf Information. 2000, (PDF ; 1,61 MB _ )

as well as the audio book CD Descartes' Traum, a compilation of his short lectures read by himself. 2002, (ISBN

3-932513-28-2)

Otto Rossler 639

References

[1] http://cnls.lanl.gov/External/

[2] Richard Gray, Science Correspondent (2010-04-28). "Legal bid to stop CERN atom smasher from 'destroying the world'" (http://www.

telegraph. co. uk/news/worldnews/europe/2650665/Legal-bid-to-stop-CERN-atom-smasher-from-destroying-the-world. html). Daily

Telegraph. . Retrieved 2008-08-30.
[3] http://www.wissensnavigator.com/download/mediumdeswissens/medium_des_wissens.pdf

External links

• Otto Rossler (http://www.uni-tuebingen.de/Chemie/Chemie/PC/Profs/roessler.html). Institut fur
Physikalische und Theoretische Chemie, Universitat Tubingen.

• Otto Rossler (http://www.atomosyd.net/spip.php7article6): From the origin of life to the architecture of chaos.
(20 October 2004). Analyse Topologique et Mo delis ation de Systemes Dynamiques.

Paul Koebe

640

Paul Koebe

Born

February 15, 1882Luckenwalde

Died

August 6, 1945 (aged 63)

Nationality

^M Germany

Fields

Mathematics

Institutions

University of Leipzig

University of Jena

Alma mater

University of Berlin

Academic advisors

Hermann Schwarz

Friedrich Schottky

Notable students

Alfred Fischer

Karl Georgi

Georg Feigl

C. Herbert Grotzsch

Ernst Graeser

Walter Brodel

Jaroslav Tagamlitski

Known for

Koebe function

Koebe 1/4 theorem

Notable awards

Ackermann— Teubner Memorial Award (1922)

Paul Koebe (February 15, 1882, Luckenwalde, Brandenburg — August 6, 1945) was a 20th-century German
mathematician. His work dealt exclusively with the complex numbers, his most important results being on the
uniformization of Riemann surfaces in a series of four papers in 1907—1909. He did his thesis at Berlin, where he
worked under Herman Schwarz. He was an extraordinary professor at Leipzig from 1910 to 1914, then an ordinary
professor at the University of Jena before returning to Leipzig in 1926 as an ordinary professor. He died in Leipzig.

Awards

• 1922, Ackermann— Teubner Memorial Award

References

[1] "Notes" (http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_l&handle=euclid.bams/

1183485532). Bulletin of the American Mathematical Society (Providence, Rhode Island: American Mathematical Society) 29 (5): p. 235.
May 1923. doi:10.1090/S0002-9904-1923-03715-4. .

External links

• Paul Koebe (http://www. genealogy. ams.org/id.php?id=19497) at the Mathematics Genealogy Project

• O'Connor, John J.; Robertson, Edmund F., "Paul Koebe" (http://www-history.mcs.st-andrews.ac.uk/
Biographies/Koebe.html), MacTutor History of Mathematics archive, University of St Andrews.

Jakob Nielsen (mathematician)

641

Jakob Nielsen (mathematician)

For other people with similar names see Jakob Nielsen.

Jakob Nielsen (October 15, 1890, Mjels-August 3,
1959, Helsing0r) was a Danish mathematician known
for his work on automorphisms of surfaces. He was
born in the village Mjels on the island of Als in North
Schleswig, in modern day Denmark. His mother died
when he was 3, and in 1900 he went to live with his
aunt and was enrolled in the Realgymnasium. In 1907
he was expelled for membership to an illicit student
club. Nevertheless, he matriculated at the University of
Kiel in 1908.

Nielsen completed his doctoral dissertation in 1913.
Soon thereafter, he was drafted into the German
Imperial Navy. He was assigned to coastal defense. In
1915 he was sent to Constantinople as a military
adviser to the Turkish Government. After the war, in
the spring of 1919, Nielsen married Carola von
Pieverling, a German medical doctor.

In 1920 Nielsen took a position at the Technical

University of Breslau. The next year he published a

paper in Mathematisk Tidsskrift in which he proved

that any subgroup of a finitely generated free group is

free. In 1926 Otto Schreier would generalize this result

by removing the condition that the free group be

finitely generated; this result is now known as the Nielsen— Schreier theorem. Also in 1921 Nielsen moved to the

Royal Veterinary and Agricultural University in Copenhagen, where he would stay until 1925, when he moved to the

Technical University in Copenhagen. He also proved the Dehn— Nielsen theorem on mapping class groups.

During World War II some efforts were made to bring Nielsen to the United States as it was feared that he would be
assaulted by the Nazis. Nielsen would, in fact, stay in Denmark during the war without being harassed by the Nazis.

In 1951 Nielsen became professor of mathematics at the University of Copenhagen, taking the position vacated by
the death of Harald Bohr. He resigned this position in 1955 because of his international undertakings, in particular
with UNESCO, where he served on the executive board from 1952 to 1958.

Jakob Nielsen

Jakob Nielsen (mathematician) 642

Bibliography

• Fenchel, Werner; Nielsen, Jakob; edited by Asmus L. Schmidt (2003). Discontinuous groups of isometries in the
hyperbolic plane. De Gruyter Studies in mathematics. 29. Berlin: Walter de Gruyter & Co..

• Nielsen, Jakob (1986), Hansen, Vagn Lundsgaard, ed., Jakob Nielsen: collected mathematical papers. Vol. 1 ,

Contemporary Mathematicians, Boston, MA: Birkhauser Boston, ISBN 978-0-8176-3140-6, MR865335

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• Nielsen, Jakob (1986), Hansen, Vagn Lundsgaard, ed., Jakob Nielsen: collected mathematical papers. Vol. 2 ,

Contemporary Mathematicians, Boston, MA: Birkhauser Boston, ISBN 978-0-8176-3151-2, MR865336

External links

• O'Connor, John J.; Robertson, Edmund F., "Jakob Nielsen (mathematician)" , MacTutor History of

Mathematics archive, University of St Andrews.

• Jakob Nielsen (mathematician) at the Mathematics Genealogy Project

References

[1] http://books.google.com/books?id=RQbvAAAAMAAJ

[2] http://books.google.com/books?id=UFKRQwAACAAJ

[3] http://www-history.mcs.st-andrews.ac.uk/Biographies/Nielsen_Jakob.html

[4] http://www. genealogy. ams.org/id.php?id=59402

Benoit Mandelbrot

643

Benoit Mandelbrot

Born

Institutions

Alma mater

Known for

Mandelbrot in 2007
20 November 1 924Warsaw, Poland

Died 14 October 2010 (aged 85)Cambridge, Massachusetts, United States

Residence Poland; France; United States

Nationality Polish, French, American

Fields Mathematics, Aerodynamics

Yale University

International Business Machines(IBM)

Pacific Northwest National Laboratory

Ecole Polytechnique

California Institute of Technology

University of Paris

Mandelbrot set, Fractals, Chaos Theory, Zipf— Mandelbrot law

Notable awards Wolf Prize (1993)
Japan Prize (2003)
The Franklin Medal
Legion d'honneur (the Legion of Honour)

Benoit B. Mandelbrot (20 November 1924 — 14 October 2010) was a French American mathematician. Born

in Poland, he moved to France with his family when he was a child. Mandelbrot spent much of his life living and
working in the United States, acquiring dual French and American citizenship.

Mandelbrot worked on a wide range of mathematical problems, including mathematical physics and quantitative
finance, but is best known as the father of fractal geometry. He coined the term fractal and described the Mandelbrot
set. Mandelbrot extensively popularized his work, writing books and giving lectures aimed at the general public.

Mandelbrot spent most of his career at IBM's Thomas J. Watson Research Center, and was appointed as an IBM
Fellow. He later became a Sterling Professor of Mathematical Sciences at Yale University. Mandelbrot also held
positions at the Pacific Northwest National Laboratory, Universite Lille Nord de France, Institute for Advanced
Study and Centre National de la Recherche Scientifique.

Benoit Mandelbrot

644

Early years

[3]

Mandelbrot was born in Warsaw into a Jewish family from Lithuania. Mandelbrot was born into a family with a
strong academic tradition - his mother was a physician and he was introduced to mathematics by two of his uncles,
one of whom, Szolem Mandelbrojt, was a mathematician who resided in Paris. However, his father made his living
trading clothing. Anticipating the threat posed by Nazi Germany, the family fled from Poland to France in 1936
when he was 11. Mandelbrot attended the Lycee Rolin in Paris until the start of World War II, when his family
moved to Tulle, France. He was helped by Rabbi David Feuerwerker, the Rabbi of Brive-la-Gaillarde, to continue
his studies. In 1944 he returned to Paris. He studied at the Lycee du Pare in Lyon and in 1945 - 47 attended the
Ecole Polytechnique, where he studied under Gaston Julia and Paul Levy. From 1947 to 1949 he studied at
California Institute of Technology, where he earned a master's degree in aeronautics. Returning to France, he

obtained his Ph.D. degree in Mathematical Sciences at the University of Paris in 1952.

From 1949 to 1958 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this
time he spent a year at the Institute for Advanced Study in Princeton, New Jersey, where he was sponsored by John

von Neumann. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland, and later to the Universite Lille

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Nord de France. In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the

T71
IBM Thomas J. Watson Research Center in Yorktown Heights, New York. He remained at IBM for 35 years,

becoming an IBM Fellow, and later Fellow Emeritus

[4]

Research career

From 1951 onward, Mandelbrot worked on problems and published
papers not only in mathematics but in applied fields such as
information theory, economics, and fluid dynamics. He became
convinced that two key themes, fat tails and self-similar structure, ran
through a multitude of problems encountered in those fields.

Mandelbrot found that price changes in financial markets did not
follow a Gaussian distribution, but rather Levy stable distributions
having theoretically infinite variance. He found, for example, that
cotton prices followed a Levy stable distribution with parameter a
equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable"
distributions have the property that the sum of many instances of a
random variable follows the same distribution but with a larger scale
parameter.

Mandelbrot also put his ideas to work in cosmology. He offered in

1974 a new explanation of Olbers' paradox (the "dark night sky"

riddle), demonstrating the consequences of fractal theory as a

sufficient, but not necessary, resolution of the paradox. He postulated

that if the stars in the universe were fractally distributed (for example,

like Cantor dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would

not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred

Mandelbrot speaking in 2007

[9]

In 1975, Mandelbrot coined the term fractal to describe these structures, and published his ideas in Les objets
fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, Chance and Dimension was
published in 1977). Mandelbrot developed here ideas from the article Deux types fondamentaux de distribution
statistique (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer,
demographer and statistician Jaromir Korcak.

Benoit Mandelbrot

645

The Mandelbrot set and periodicities of orbits.

While on secondment as Visiting Professor of Mathematics at Harvard
University in 1979, Mandelbrot began to study fractals called Julia sets
that were invariant under certain transformations of the complex plane.
Building on previous work by Gaston Julia and Pierre Fatou,
Mandelbrot used a computer to plot images of the Julia sets of the

formula z - |x. While investigating how the topology of these Julia
sets depended on the complex parameter u. he studied the Mandelbrot
set fractal that is now named after him. (Note that the Mandelbrot set is

now usually defined in terms of the formula z + c, so Mandelbrot's
early plots in terms of the earlier parameter \i are left— right mirror
images of more recent plots in terms of the parameter c.)

ri2i

In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature. This influential work
brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had
dismissed fractals as "program artifacts".

Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM

ri3i

decided to end pure research in his division. He joined the

Department of Mathematics at Yale, and obtained his first tenured post

ri4i
in 1999, at the age of 75. At the time of his retirement in 2005, he

was Sterling Professor of Mathematical Sciences. His awards include

the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of

the European Geophysical Society in 2000, the Japan Prize in 2003,

and the Einstein Lectureship of the American Mathematical Society in

2006.

Mandelbrot speaking about the Mandelbrot set,

during his acceptance speech for the Legion

d'honneur in 2006

The small asteroid 27500 Mandelbrot was named in his honor. In

November 1990, he was made a Knight in the French Legion of

Honour. In December 2005, Mandelbrot was appointed to the position

of Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was promoted to Officer of the

Legion of Honour in January 2006. An honorary degree from Johns Hopkins University was bestowed on

Mandelbrot in the May 2010 commencement exercises

[17]

Fractals and regular roughness

Although Mandelbrot coined the term fractal, some of the
mathematical objects he presented in The Fractal Geometry of Nature
had been described by other mathematicians. Before Mandelbrot, they
had been regarded as isolated curiosities with unnatural and
non-intuitive properties. Mandelbrot brought these objects together for
the first time and turned them into essential tools for the long-stalled
effort to extend the scope of science to non-smooth objects in the real
world. He highlighted their common properties, such as self-similarity
(linear, non-linear, or statistical), scale invariance, and a (usually)
non-integer Hausdorff dimension.

A limb of a maple tree, illustrating organic fractal
branching.

He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world.
Natural fractals include

Benoit Mandelbrot

646

the shapes of mountains, coastlines and river basins; the structures of
plants, blood vessels and lungs; the clustering of galaxies; and
Brownian motion. Fractals are found in human pursuits, such as music,
painting, architecture, and stock market prices. Mandelbrot believed
that fractals, far from being unnatural, were in many ways more
intuitive and natural than the artificially smooth objects of traditional
Euclidean geometry:

Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor does
lightning travel in a straight line.
— Mandelbrot, in his introduction to The Fractal
Geometry of Nature

Natural water frost crystal growth on cold glass,

showing fractal branching growth in a purely

physical system.

Mandelbrot has been called a visionary and a maverick. His informal and passionate style of writing and his
emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal
Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and
contributed to chaos theory and other fields of science and mathematics.

When visiting the Museu de la Ciencia de Barcelona in 1988, he told its director that the painting The Face of War
had given him "the intuition about the transcendence of the fractal geometry when making intelligible the
omnipresent similitude in the forms of nature". He also said that, fractally, Gaudi was superior to Van der

Rohe

[20]

Death

Mandelbrot died in a hospice in Cambridge, Massachusetts, on 14 October 2010 from pancreatic cancer, at the age of

[21] [221

85. Reacting to news of his death, mathematician Heinz-Otto Peitgen said "if we talk about impact inside

1211

mathematics, and applications in the sciences, he is one of the most important figures of the last 50 years." Chris

T231
Anderson described Mandelbrot as "an icon who changed how we see the world." French President Nicolas

Sarkozy said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering

preconceived notions". Sarkozy also added, "His work, developed entirely outside mainstream research, led to

1241

modern information theory." Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond

the academy" and lauds him as the "father of fractal geometry

.,[25]

Honors and awards

\ r )fA

A partial list of awards received by Mandelbrot:

Benoit Mandelbrot

647

2004 Best Business Book of the Year

Award

AMS Einstein Lectureship

Barnard Medal

Caltech Service

Casimir Funk Natural Sciences Award
Charles Proteus Steinmetz Medal
Franklin Medal

Harvey Prize

Honda Prize

Humboldt Preis

Fellow, American Geophysical

Union

IBM Fellowship

Japan Prize

John Scott Award

Legion d'honneur (Legion of

Honour)

Lewis Fry Richardson Medal

Medaglia della Presidenza della Repubblica

Italiana

Medaille de Vermeil de la Ville de Paris

Nevada Prize

Science for Art

Sven Berggren-Priset

Wtadyslaw Orlicz Prize

Wolf Foundation Prize for Physics

Member of the Norwegian Academy of Science and Letters

[27]

Notes

[I] Mandelbrot chose his own middle initial, but it doesn't stand for anythingLesmoir-Gordon, Nigel (17 October 2010). "Benoit Mandelbrot
obituary" (http://www.guardian.co.uk/science/2010/oct/17/benoit-mandelbrot-obituary). The Guardian. . Retrieved 17 October 2010.

[2] Pronounced English pronunciation: /'maendalcbrDt/ MAN-d3l-brot in English." Mandelbrot (http://oed.com/

search?searchType=dictionary&q=Mandelbrot)". Oxford English Dictionary. Oxford University Press. 2nd ed. 1989. When speaking in

French, Mandelbrot pronounced his name [benwa mddSlbKot]. Recording of the September 1 1, 2006, ceremony at which Mandelbrot received

the Officer of the Legion of Honour insignia
[3] Mandelbrot, Benoit; Bernard Sapoval, Daniel Zajdenweber (May 1998). "Web of Stories • Benoit Mandelbrot • Family background and early

education" (http://www.webofstories.com/play/9596). Web of Stories. . Retrieved 19 October 2010.
[4] Mandelbrot, Benoit (2002). "A maverick's apprenticeship" (http://www.math.yale.edu/mandelbrot/web_pdfs/mavericksApprenticeship.

pdf). The Wolf Prizes for Physics. Imperial College Press. .
[5] "BBC News - 'Fractal' mathematician Benoit Mandelbrot dies aged 85" (http://www.bbc.co.uk/news/world-europe-l 1560101). BBC

Online. 17 October 2010. . Retrieved 17 October 2010.
[6] Hemenway P. Divine proportion: Phi in art, nature and science. Psychology Press, 2005 ISBN 0415344956
[7] Barcellos, Anthony (1984). "Interview Of B. B. Mandelbrot" (http://www.math.yale.edu/mandelbrot/web_pdfs/inHisOwnWords.pdf).

Mathematical People. Birkhailser. .
[8] ""New Scientist", 19 April 1997" (http://www.newscientist.com/article/mgl5420784.700-flight-over-wall-st.html). Newscientist.com.

1997-04-19. . Retrieved 2010-10-17.
[9] Galaxy Map Hints at Fractal Universe, by Amanda Gefter; New Scientist; June 25, 2008
[10] Fractals: Form, Chance and Dimension, by Benoit Mandelbrot; W H Freeman and Co, 1977; ISBN 0716704730

[II] Jaromi'r Korcak (1938): Deux types fondamentaux de distribution statistique. Prague, Comite d'organisation, Bull, de l'Institute Int'l de
Statistique, vol. 3, pp. 295-299.

[12] The Fractal Geometry of Nature (http://books. google. com/books?id=xJ4qiEBNP4gC&printsec=frontcover), by Benoit Mandelbrot; W H

Freeman & Co, 1982; ISBN 0716711869
[13] Mandelbrot, Benoit; Bernard Sapoval, Daniel Zajdenweber (May 1998). "Web of Stories • Benoit Mandelbrot • IBM: background and

policies" (http://www.webofstories.com/play/10483). Web of Stories. . Retrieved 17 October 2010.
[14] Tenner, Edward (16 October 2010). "Benoit Mandelbrot the Maverick, 1924-2010" (http://www.theatlantic.com/technology/archive/

2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/). The Atlantic. . Retrieved 16 October 2010.
[15] "PNNL press release: Mandelbrot joins Pacific Northwest National Laboratory" (http://www.pnl. gov/news/release.asp?id=141).

Pnl.gov. 2006-02-16. . Retrieved 2010-10-17.
[16] ""Legion d'honneur" announcement of promotion of Mandelbrot to "officier"" (http://www.legifrance.gouv.fr/WAspad/

UnTexteDeJorf?numjo=PREX0508911D) (in (French)). Legifrance.gouv.fr. . Retrieved 2010-10-17.
[17] "Six granted honorary degrees, Society of Scholars inductees recognized" (http://gazette.jhu.edu/2010/06/07/

six-granted-honorary-degrees-society-of-scholars-inductees-recognized-2/). Gazette.jhu.edu. 2010-06-07. . Retrieved 2010-10-17.
[18] Devaney, Robert L. (2004). ""Mandelbrot's Vision for Mathematics" in Proceedings of Symposia in Pure Mathematics. Volume 72.1" (http:/

/www. math. yale.edu/mandelbrot/web_pdfs/jubileeletters. pdf). American Mathematical Society. . Retrieved 2007-01-05.
[19] Jersey, Bill (April 24, 2005). "A Radical Mind" (http://www.pbs.org/wgbh/nova/fractals/mandelbrot.html). Hunting the Hidden

Dimension. NOVA/ PBS. . Retrieved 2009-08-20.
[20] Jorge Wagensberg (20 October 2010) (in Spanish). Obituarios. El descuhridor de fractales. p. 37
[21] Hoffman, Jascha (16 October 2010). "Benoit Mandelbrot, Mathematician, Dies at 85" (http://www.nytimes.com/2010/10/17/us/

17mandelbrot.html). The New York Times. . Retrieved 16 October 2010.

Benoit Mandelbrot 648

[22] "Benoit Mandelbrot, fractals pioneer, dies" (http://www.upi.com/Science_News/2010/10/16/Benoit-Mandelbrot-fractals-pioneer-dies/

UPI-11551287266964/). United Press International. 16 October 2010. .Retrieved 17 October 2010.
[23] "Mandelbrot, father of fractal geometry, dies" (http://www.montrealgazette.com/technology/Mandelbrot-l-father-l-fractal-l-geometry-l-

dies/3682961/story.html). The Gazette. . Retrieved 16 October 2010.
[24] "Sarkozy rend hommage a Mandelbrot [Sarkozy pays homage to Mandelbrot]" (http://www.lefigaro.fr/flash-actu/2010/10/16/

97001-20101016FILWWW0061 1-sarkozy-rend-hommage-a-mandelbrot.php) (in French), he Figaro. . Retrieved 17 October 2010.
[25] Benoit Mandelbrot's obituary (The Economist) (http://www.economist.com/node/17305197)
[26] Mandelbrot, Benoit B. (2 February 2006). "Vita and Awards (Word document)" (http://www.math.yale.edu/mandelbrot/web_docs/

VitaSeveralPage.doc). . Retrieved 2007-01-06.
[27] "Gruppe 1: Matematiske fag" (http://www. dnva.no/c26849/artikkel/vis. html?tid=401 16) (in Norwegian). Norwegian Academy of

Science and Letters. . Retrieved 7 October 2010.

References
Further reading

• Hudson, Richard L.; Mandelbrot, Benoit B. (2004). The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin,
and Reward. New York: Basic Books. ISBN 0-465-04355-0.

• Mandelbrot, Benoit; Taleb, Nassim (23 March 2006). "A focus on the exceptions that prove the rule" (http://
w w w . ft . com/cms/s/2/

5372968a-ba82-llda-980d-0000779e2340,dwp_uuid=77a9a0e8-b442-l Ida-bd61-0000779e2340.html).
Financial Times. Retrieved 2010-10-17.

• Heinz-Otto Peitgen, Hartmut Jiirgens, Dietmar Saupe and Cornelia Zahlten: Fractals: An Animated Discussion
(63 min video film, interviews with Benoit Mandelbrot and Edward Lorenz, computer animations), W.H.
Freeman and Company, 1990. ISBN 0-716-72213-5 (re-published by Films for the Humanities & Sciences, ISBN
978-0-7365-0520-8)

• Mandelbrot, Benoit (February 1999). "A Multifractal Walk down Wall Street". Scientific American.

• "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and
deepening our understanding of nature" (http://www.pbs.org/wgbh/nova/fractals/), NOVA, WGBH TV,
PBS, October 28, 2008.

• Mandelbrot, Benoit B.; Freeman, W. H. (1983). The Fractal Geometry of Nature. San Francisco: W.H. Freeman.
ISBN 0-7167-1186-9.

External links

• Mandelbrot's page at Yale (http://www.math.yale.edu/mandelbrot/)

• Ted talk: "Benoit Mandelbrot: Fractals and the art of roughness" (http://www.ted.com/talks/
benoit_mandelbrot_fractals_the_art_of_roughnes s . html)

• Interview at FT.com (http://video.ft.eom/v/63078298001/Why-efficient-markets-collapse-Mandelbrot), in
which Mandelbrot discusses his early work with stock market behavior, 2009-09-30

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Computational physics Source: http://e11.wikipedia.0rg/w/index. php?oldid=420667192 Contributors: APH, Andrewwall, Bduke, Berg0747, Burn, CBM, Chaos, CyborgTosser, D6, David
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Mathematical Formulations of Quantum Dynamics Source: http://en.wikipedia.org/w/index.php?oldid=4241 00657 Contributors: 2over0, 41ex, Akriasas, Alaexis, Ancheta Wis, Andre Engels,
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Quantum Chemistry and Biochemical Dynamics Source: http: //en. wiki pedia.org/ w/index.php?oldid=4 209 16611 Contributors: 144.1 89.40.xxx, 208.40. 185.xxx, 41ex, AManWithNoPlan,
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List of quantum chemistry and solid state physics software Source: http://en.wikipedia.org/w/index.php?oldid=424767977 Contributors: Aeleen, Bduke, Bob K31416, Colonies Chris,
Cudauser, Enric Naval, Ergowikiuser, Felixhanke, Gerrit C. Groenenboom, Giftlite, Gioto, Gxf57, Hbarlab, JLaTondre, JRmoret, JaGa, Jeff.science, Keenan Pepper, Laudak, Mdt26a, Michael
Hardy, Mipmip, NNemec, Nicolasbock, Nilox83, Novel compound, P99am, Ph0987, Qrage27, Radovan.bast, RosaWeber, Rpmuller, Rwwww, THEN WHO WAS PHONE?, Tdoune, Vblum,
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Sequential dynamical system Source: http://en.wikipedia.org/w/index.php?oldid=372568536 Contributors: CBM, Charvest, Delaszk, Giftlite, Michael Hardy, Oconnor663, Wikimathman, 3
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Automata theory Source: http://en.wikipedia.Org/w/index. php'?oldid=423428720 Contributors: A Generic Reality, Aaron Schulz, Ahmad. shahw an, Ahoerstemeier, Alansohn, Allan Mclnnes,
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Time series analysis Source: http://en.wikipedia.org/w/index.php?oldid=16085058 Contributors: Charles Matthews, Michael Hardy, PAR, Rgclegg, Taxman, Wile E. Heresiarch, Wyllium, 3
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Lag operator Source: http://en.wikipedia.org/w/index.php?oldid=42384701 1 Contributors: Albmont, Arfhena, Bequw, J heisenberg, Keilana, Ludovic89, Melcombe, Michael Hardy, Oli Filth,
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Shift operator Source: http://en.wikipedia.org/w/index.php?oldid=396637821 Contributors: Albmont, Archelon, Arienh4, BenFrantzDale, CBM, Charles Matthews, Linas, Lupin, Magioladitis,
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Shift space Source: http://en.wikipedia.0L'g/\v/indcx.php Void id=-4 2-19-143 1 I Contributors: Charvest, Fridemar, Fudo, Giftlite, Lantonov, Rjwilmsi, SiamakT, Trovatore, Wpathooper, 1
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Markov partition Source: http://en.wikipedia.org/w/index.php?oldid=414949277 Contributors: Arcfrk, Charles Matthews, DavidCBryant, GregorB, J12Tap, JHunterJ, Linas, Melcombe, 2
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Sharkovskii's theorem Source: http: //en. wikipedia.org/w /index. php'?oldid=4 22874747 Contributors: Algebraist, Andriyko, AxelBoldt, Badpazzword, BeteNoir, Catslash, Charles Matthews,
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Ergodic system Source: http://e11.wikipedia.0rg/w/index. php?oldid=22486694 Contributors: Charles Matthews, D6, Dcljr, Dysprosia, Fredrik, Klaus scheicher, Linas, Michael Hardy, Ojigiri,
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Ergodic theory Source: http://en.wikipedia.org/w/index.php'?oldid=424491 160 Contributors: Charles Matthews, D6, Dcljr, Dysprosia, Fredrik, Klaus scheicher, Linas, Michael Hardy, Ojigiri,
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Measure-preserving dynamical system Source: http://en.wikipedia.org/w/index.php?oldid=4 11488446 Contributors: Alai, Arcfrk, Blh3321, Charles Matthews, Cmk5b, Erzbischof, False
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Periodic orbit Source: http://en.wikipedia.org/w/index.php?oldid=l 1 1580721 Contributors: AxelBoldt, Charles Matthews, Fropuff, Hawthorn, Isomorphic, Linas, Lupin, Mat cross,
MathMartin, Patrick, Sauve.d, That Guy, From That Show!, The Anome, Tosha, XaosBits, 1 anonymous edits

Article Sources and Contributors 652

Hilbert space Source: http://en.wikipedia.org/w/index.php?oldid=424904266 Contributors: 128.95.173.xxx, Idiot, Abecedare, Aetheling, Anville, Aram33, Arcfrk, Archelon, Arthur Rubin,
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Spherical harmonics Source: http://en.wikipedia.org/wAndex. php?oldid=425048939 Contributors: 0.39, Acolombi, Alain Michaud, Albmont, Aldaron, Almit39, Awickert, Axiomsofchoice,
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Quantum computer Source: http://en.wikipedia.org/w/index.php?oldid=425785187 Contributors: -Ozone-, 194.117.133.xxx, lmujin22, 41ex, A5b, AAAAA, AWeishaupt, Aarchiba,
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Topological Quantum Computers Source: http://en.wikipedia.org/w/index.php?oldid=416402781 Contributors: Crescent 1, Gil987, Harold f, Herbee, Ihopel27, Jeffq, Jkasd, Justin545,
Korepin, Lostart, Maayanh, Materialscientist, Michael Hardy, Pwjb, RobinK, Shlomi Hillel, Sizzle Flambe, Swerve2310, Slawomir Bialy, Teply, That Guy, From That Show!, Wireader,
Woottonjames, 26 anonymous edits

Algebraic Geometry Source: http://en.wikipedia.org/w/index.php?oldid=424756955 Contributors: 128.252.121.xxx, 171.64.38. xxx, APH, Acepectif, After Midnight, Alansohn, Alsandro,
Andrei Polyanin, Anonymous Dissident, Antandrus, Arcfrk, AxelBoldt, Bachrach44, Bender235, BigFatBuddha, Bo Jacoby, Bobblewik, Bryan Derksen, C questOOO, CBM, CRGreathouse,
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Category theory Source: http://en.wikipedia.org/w/index.php?oldid=4221 12233 Contributors: 0, 63. 162. 153. xxx, 7.239, APH, Adrianwn, Alexwright, Alidev, Alriode, Anonymous Dissident,
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Graph dynamical system Source: http://en.wikipedia.org/w/index.php?oldid=418085477 Contributors: Charvest, Docu, Eslipl7, Harryboyles, Henning.Mortveit, Jim.belk, Jyoshimi, Mhym,
Michael Hardy, 3 anonymous edits

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Analysis of Systems Source: http://en.wikipedia.org/w/index.php?oldid=416144640 Contributors: Ace Frahm, Barneca, BeenBeren, Bestestngineer, Cburnett, Chimmychimp, Cirrus 1,
CyborgTosser, Cyrus Grisham, Derek Ross, Dicklyon, DoriSmith, Geoeg, Jayman3000, Jpbowen, Jujutacular, Mark Renier, Mbell, McGeddon, Mdd, Michael Devore, Mkweise, Mrdthree,
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Dynamic Bayesian network Source: http://en.wikipedia.org/w/index.php?oldid=376335525 Contributors: Arodichevski, Charles Matthews, MarkWahl, Morgaladh, Obankston, Tomixdf, Zeno
Gantner, 3 anonymous edits

Dynamic network analysis Source: http://en.wikipedia.org/w/index.php?oldid=3924855 1 Contributors: 477 TalaB. AZK. AbsolutDan. Ace Frahm. Argon233. Beetslra, Delaszk, Diabloblue.
Douglas R. White, Erkan Yilmaz, Imersion, JonHarder, Lauyukpui, Mattdereno, Melcombe, Michael Hardy, Mindl23, Porqin, SiobhanHansa, Supernet, Supertabular, Terrillfrantz, Wikidemon,
Yassens, 18 anonymous edits

Dynamic circuit network Source: http://en.wikipedia.org/w/index.php?oldid=397140678 Contributors: Kbrose, Rjwilmsi, 3 anonymous edits

Tensor product network Source: http://en.wikipedia.org/w/index.php?oldid=413066012 Contributors: Bci2, CharlesGillingham, Dysprosia, Foobarnix, JonHarder, Oracleofottawa, R'n'B, Saga
City

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Langner, Kelisi, Kenneth M Burke, Kesaloma, Kevin Baas, LSmok3, Larry laptop, Lexor, Lightmouse, LilHelpa, Lolzolzolz, Loremaster, Lowellian, Luctor IV, M.e, Magnusr, ManuelGR,
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Veritas, Vlad, Vuo, Warrickball, Wikiborg, Will Beback, Wotnow, Xgeom, Zoniedude, AjieKcaHAtp, 278 anonymous edits

Systems Biology Source: http://en.wikipedia.org/w/index.php?oldid=425637491 Contributors: APH, Aciel, Alan Liefting, AlirezaShaneh, Amaher, Amandadawnbesemer, Amirsnik, Andreas td,
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Neurosciences Source: http://en.wikipedia.org/w/index.php?oldid=423970546 Contributors: .seVer!Ty A -, 168..., 16@r, 3rdAlcove, 62.2.17.xxx, A314268, AHands, APH, Aaron Schulz,
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Biocybernetics Source: http://en.wikipedia.org/w/index.php?oldid=425659771 Contributors: Beren, DanielNuyu, Gogo Dodo, Joegoodbud, Joerom5883, Mdd, Oldekop, Oxymoron83,
Scorpion451, Shawnc, Shureg, Slicky, Stemonitis, Tarcieri, Voldemore, Xanzzibar, 21 anonymous edits

Computational neuroscience Source: http://en.wikipedia.org/w/index.php?oldid=422241445 Contributors: A314268, APH, ASmartKid, Aaron Schulz, Alex. tan, Alterego, Alv, Andorin,
Ariangiovanni, Arne Heise, Ashleyisachild, Bayle Shanks, Benhocking, BertSeghers, Brainboychik, Cardoso44, Cardosodeoliveira, CasperEACClemence, CesarB, Ceyockey, Chardar, Chealer,
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Molecular Neurosciences and Molecular Medicine Source: http://en.wikipedia.org/w/index.php?oldid=423971 124 Contributors: Clicketyclack, Danielkueh, Gurch, Jonkerz, Kmd024000,
Rich Farmbrough, Rjwilmsi, Shushruth, Sluox, SriMesh, TenOfAllTrades, Tryptofish, 2 anonymous edits

Complex Systems Source: http://e11.wikipedia.0rg/w/index. php?oldid=42 1528049 Contributors: &Delta, 130. 94.122. xxx, Acadac, Ace Frahm, Adam M. Gadomski, AdamRetchless, Adastra,
Amalas, Antoni Barau, Apofhecia, Ballista, Betacommand, Betamod, Bryan Derksen, Calltech, Cazort, Centrx, Childhoodsend, Choesarian, Chopchopwhitey, Christopher Agnew, Cmanske,
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Complex Systems Biology Source: http://en.wikipedia.org/wAndex. php?oldid=408273124 Contributors: Bci2, Dolovis, Michael Hardy, Rjwilmsi, 1 anonymous edits

Mathematical, Relational and Theoretical Biology Source: http://en.wikipedia.org/w/index.php?oldid=425644706 Contributors: Adoniscik, Agilemolecule, Agricola44, Alan Liefting,
Anclation, Andreas td, Aua, Audriusa, Baz. 77. 243. 99. 32, Bci2, Bduke, Bender235, Berland, BillWSmithJr, Ceyockey, Charvest, Chopchopwhitey, Chris Capoccia, Commander Nemet,
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Tpbradbury, Tremilux, Triwbe, Vina, Wavelength, Way sTo Escape, Wpoeop, 99 anonymous edits

Lotka-Volterra Differential Equations in Population Biology Source: http: //en. wiki pedia.org/ w/index.php?oldid=4 24666076 Contributors: Adiel lo, Alan Toomre, Alansohn, AllenDowney,
Arthur Rubin, B5Jaeger, Berland, BigJohnHenry, Billjefferys, Bryan Derksen, CALR, Charles Matthews, Chochopk, Chopchopwhitey, Commander Nemet, Conti, Cutler, Cyborg Tosser,
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Chaos theory Source: http://en.wikipedia.org/w/index.php?oldid=425474335 Contributors: 19.168, 19.7, 3pl416, 4dhayman, 7h3 3L173, Abrfreek777, Academic Challenger, Adam Bishop,
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Chaos theory in organizational development Source: http://en.wikipedia.org/w/index.php?oldid=406615536 Contributors: 9Nak, Bookgrrl, BradlOl, El C, Gobbleswoggler, IPSOS,
Kotinopoulos, Letranova, Pedroch, Rjwilmsi, Rkelly74, Schroep, 15 anonymous edits

Attractor Source: http://en.wikipedia.org/w/index.php?oldid=424363150 Contributors: Akella, AndrewKepert, Bergsten, Booyabazooka, BryanD, Charles Matthews, Daqu, Deeptrivia,
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Lorenz attractor Source: http://en.wikipedia.org/w/index.php?oldid=425372437 Contributors: 0, Academic Challenger, Ahoerstemeier, Anonymous Dissident, Antonio Monreal Acosta,
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Tommynoddy, To campos, Wikimol, Yill577, Zlobny, 91 anonymous edits

Strange attractor Source: http://en.wikipedia.org/w/index.php?oldid= 173578964 Contributors: Akella, AndrewKepert, Bergsten, Booyabazooka, BryanD, Charles Matthews, Daqu, Deeptrivia,
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Butterfly effect Source: http://en.wikipedia.org/w/index.php?oldid=425672567 Contributors: lOmetreh, 2D, 2QT2BSTR8, 2over0, A Quest For Knowledge, A dullard, A. Parrot, Aitias,
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Article Sources and Contributors 655

Themastertree, Thomas G Graf, Thumperward, Toad of Steel, Tommy2010, Tony Sidaway, Trovatore, To campos, Undsoweiter, User2004, Uusitunnus, Valley2city, Vary, VasilievVV, Ventura,
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Standard map Source: http://en.wikipedia.org/w/index.php'?oldid=407043874 Contributors: 3pl416, Auntof6, Doroncohen, El C, Erpert, John of Reading, Linas, Michael Hardy, Pmg, Robertd,
Rychlik, Shepelyansky, Supertjhok, 6 anonymous edits

Henon map Source: http://en.wikipedia.org/w/index.php?oldid=423 175056 Contributors: Andreww, Bryan Derksen, Charles Matthews, Don Warren, EmilJ, Fractalmichel, Fredrik, Grutness,
Herbee, Isomorphic, K-UNIT, Lhf, Linas, Martin Hinks, Michael Hardy, Mr0tl633, Oleg Alexandrov, PAR, Reetep, Ritchy, Rjwilmsi, Scythe33, Silverfish, Stepa, The Anome, XaosBits, 14
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Horseshoe map Source: http://en. wikipedia. org/w/index.php'?oldid=421074601 Contributors: AmitAronovitch, Charles Matthews, DanielRigal, Darkwind, Gaius Cornelius, Jasper Chua, Jitse
Niesen, K-UNIT, Linas, Lunch, Michael Hardy, Mousomer, PhilipMW, Pokipsy76, R.e.b., Sverdrup, SyntaxError55, Sysy, Timwi, Urhixidur, William M. Connolley, XaosBits, ZooFari, 24
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Coupled map lattice Source: http://en.wikipedia.org/w/index.php?oldid=414881775 Contributors: Gadget850, Melaen, Michael Hardy, Mrocklin, Nick Number, RHaworth, Torcini, Travdog8,
Twri, 77 anonymous edits

List of chaotic maps Source: http://en.wikipedia.org/w/index.php?oldid=422973516 Contributors: Adam majewski, David spector, Duoduoduo, Guroadrunner, Hugo.cavalcante, JocK, Linas,
Louelle, Maxlittle2007, Mikesl988, Oleg Alexandrov, PAR, Plrk, Pokipsy76, Premeditated Chaos, TravdogS, Twri, ZeroOne, 23 anonymous edits

Chua's circuit Source: http://en.wikipedia.org/w/index.php?oldid=417923067 Contributors: 3pl416, Arthur Rubin, Bkell, Coredesat, Dlyons493, Etphonehome, Glenn, Jafet, Jeremyberman,
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Double pendulum Source: http://en.wikipedia.org/w/index.php?oldid=425 187680 Contributors: 19.7, Acdx, Alias Flood, Andres, ArnoldReinhold, Berland, Brad, Brian0918, Bvanderveen,
CALR, Caltas, Catslash, Cburnett, Cdfurman, Chetvorno, Chopchopwhitey, Cimon Avaro, CommonsDelinker, Conscious, Doradus, Grendelkhan, Hughhunt, Iridescent, JabberWok, Jagl23,
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Thing That Should Not Be, Themfromspace, Thurth, Timb66, Tom harrison, Ummit, Why My Fleece?, Wolfkeeper, XaosBits, Xenvir, Zakholdsworfh, 77 anonymous edits

Dynamical billiards Source : http://en.wikipedia.org/w/index.php?oldid=414919077 Contributors: AllynJ, Arcfrk, Bleeve, CALR, Charles Matthews, Chris the speller, Chuunen Baka,
Cnmirose, CustosO, Dbezanson, Domitori, Doroncohen, Dtrebbien, Dyaa, GJAckland, GermanX, GregorB, Hektor, Henning Makholm, Hetar, HoosBB, Jitse Niesen, K-UNIT, KSmrq, Kelly
Martin, Korako, Linas, Lusile, Mboverload, Mdd, Melchoir, Michael Hardy, Monguin61, Oleg Alexandrov, OrangeDog, Ortvolute, Quazmapod, Rpchase, SMcCandlish, Seventy Three, Slawomir
Bialy, Tarotcards, Tepid451, Trapezoidsky, XJamRastafire, XaosBits, 33 anonymous edits

Bifurcation theory Source: http://en.wikipedia.org/w/index.php?oldid=425471 173 Contributors: Amitkashwar, Anandhan, Arthur Rubin, Athkalani, Bender235, Cat2020, Deeptrivia,
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Zsniew, 47 anonymous edits

Rossler attractor Source: http://en.wikipedia.org/w/index.php?oldid=422715218 Contributors: AndrewKepert, Art LaPella, Can't sleep, clown will eat me, Carrp, Cquimper, El C, Eteq,
Fnielsen, Freywa, GregorB, Hqb, Joakim Munkhammar, Juangamin, Kwamikagami, Logicalrealist, Mpkilla, Oleg Alexandrov, Prokofiev2, RoonytOOns, Selket, SnowRaptor, To campos, 37
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Synchronizing Chaos Source: http://en.wikipedia.org/w/index.php?oldid=403030436 Contributors: 3pl416, Battocchia, IRP, Inductiveload, Jitse Niesen, Ninly, PhilKnight, QueenCake,
Versus22, 23 anonymous edits

The Possibility of Quantum Chaos ? Source: http: //en. wikipedia. org/ w/index.php?oldid=425 608464 Contributors: Ahoerstemeier, Baxxterr, CRGreathouse, Cfailde, Charles Matthews,
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Lightmouse, Linas, Mdd, Michael Courtney, MoonMan, Neilc, Noeckel, Open2universe, Osarius, Phisquare, PhysicsBob, Pravika, Quintote, Rmrfstar, Shenme, Shepelyansky,
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Fractals and Fractional Dimensions Source: http ://en. wikipedia. org/w/index.php?oldid=425 5 74303 Contributors: 149.99.222.xxx, 16@r, 5theye, ABCD, APH, AVM, Aafuss, Abce2,
Accuruss, Acdx, Adamhauner, Aditya, AdjustShift, Ael 2, Aelffin, Aeternus, Afteread, Ahmad.829, Ahoerstemeier, Aidje, Aitias, Alansohn, AlexKepler, Andersmusician, Andrewlpl991, Andris,
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Samuel Eilenberg Source: http://en.wikipedia.org/w/index.php?oldid=421762365 Contributors: Ajank, Almit39, Avraham, Bender235, Bnynms, C S, CJLL Wright, CRKingston, Calle,
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Chocolateboy, Choster, Chowells, Chris 73, Chrisd87, Chrisdempsey, Chrislk02, Christian Giinther, Christopher Parham, ChristopherWillis, Chronicler, Chryed, Chu333222, CieloEstrellado,
CillasOOl, Cirt, Clarityfiend, Clayoquot, Cleared as filed, Clearhistory, Clemwang, ClockworkSoul, Closeapple, Clubjuggle, Cnilep, Codetiger, Codex Sinaiticus, Colbain, College Watch,
Colonel Mustard, CommonsDelinker, Complexica, Comrade Che 1, Conti, Conversion script, Cookiemobster4, Cool3, CoolKidl993, Coolgyingman, Cordless Larry, Corevette, Corinne68,
Corpx, Corti, Couki, Cousinroth, Coyets, Cpl Syx, Cradel, CrazyChemGuy, Crazycomputers, Crazylupin, Crea, Creaml47, Crito2161, Cronholml44, CryptoDerk, Csernica, Ctivh07,
Culnacreann, Curps, Curtis wagner ball, Cyan, Cyclekidl500, Cyktsui, CynicalMe, Cypriot stud, D-Rock, D.H, D6, DAGwyn, DCEdwardsl966, DCGeist, DGJM, DHN, DIREKTOR, DJ
Clayworth, DJ20005992, DJGB, DLBY, DO'Neil, DOHC Holiday, DOSGuy, DVD R W, DVdm, Daa89563, Dabbijo, Dabomb87, DadaNeem, Dan Austin, Dan Gan, DanlOO, DanKeshet,
DancingPenguin, Dancter, Danh, Daniel, Daniel 123, Daniel Collins, Daniel5127, DanielCD, Danijobi, Danny lost, Dannyno, Dantadd, Daphne A, Dark Wolf 1, Darry2385, Darth Livers, Darth
Sidious, Dave00327, DaveApter, Daven200520, Davewild, Davey9678, David Underdown, David Wallace Croft, David.Mestel, DavidWBrooks, Davidizerl3, Davshul, Dawn Bard, Dazacman,
Db09922 1 , Dbmerge, Dbtfz, Dcflyer, Dddstone, Ddihel, De kludde, Deadcorpse, Debresser, Decltype, Deepak, Deerstop, Defender of torch. Deflective, Deinocheirus, Delldot, Deltabeignet,
Demmy, Denni, Deputydog23, Deputyduck, Der Golem, DerHexer, Derek.cashman, Derfman24, Derwig, Desdinova, Dewan357, Deyyaz, Dezzo, Dfrg.msc, Dgroseth, Dhemm, Dhodges, Diberri,
Dicklyon, Diego pmc, Digidestine, Digitalme, Dimadick, Dinotitan, Disavian, Discospinster, Dismas, Dispenser, Divine Corsair, Divius, Djehowell, Djmutex, Djnjwd, Dlohcierekim's sock,
Dmackey2, Dman5555414n52, Dmcq, Dmk, Dmn, Dmoonl, Dmv74, Dncarley, Doc glasgow, DocRocksl, Doctorpete, Domitori, DonSiano, Donatus, Donfbreed, Donreed, Doomed Rasher,
Dorftrottel, Doubtfulsound, Dp074, Dp462090, Dpbsmith, Dpotop, Dr whol975, Dr. Chaos, Dr. Submilhmeter, Dr.K., DrKiernan, DragonflySixtyseven, Dramedy Tonight, Drawmelpark,
Drbogdan, Drbreznjev, Dreftymac, Drhundertwasser, Drinksquare, Drizzd, Drunken Pirate, Dryazan, Dswim279, Dtgm, Dtgriscom, Duagloth, Ducker, Dudzcom, Dugwiki, Duncharris, Dune
jumpr99, Dwslassls, Dyslexic agnostic, E-lord, E4mmacro, EEPROM Eagle, ERcheck, ESkog, EZ, Eaglesm20, EarthBoundX5, Earthlyreason, East718, Ed Brey, Ed g2s, EdJohnston, Edcolins,
Eddiesgirll 121, Eddy23, EdgarlSl, Edinborgarstefan, Edivorce, Edsmilde, Edton, Edwy, Eean, EfAston, Eirik (usurped), Ekem, El C, ElKevbo, ElTyrant, Eladeira, Eldaran, Eleuther, Elias Enoc,
ElinorD, Eljim, Ellisonch, Eloquence, Elwaystalker007, Emax, Emc2, Emerson7, Emijrp, EmilJ, Emops, Emperor Azure, Ems57fcva, Enchanter, Enigmaman, Ent, Epbrl23, Eppyie, Epsiloner,
Epson291, EronMain, Erudecorp, EscapingLife, Esolbiz, Espritl5d, Essjay, EstebanF, Esterson, Et9871, Ethan Mitchell, Eubulides, EugeneZelenko, EvanBlom, Evankordimosh, Evenfiel,
Everyking, Evil Monkey, Evolauxia, Exeunt, FF2010, FOTEMEH, Facembanorth, Fair Deal, Fanatix, Fang Aili, Fantasy, Farbish, Farooqmo, Farosdaughter, Fastfission, Favonian, FayssalF,

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Fconaway, FeanorStar7, Feezo, Feitclub, Feketekave, Fennec, Fernando S. Aldado, FestivalOfSouls, Feureau, Fgjkdfjkd, Fgnievinski, Fgoall3, Filliam, Finn-Zoltan, Firefly322, Fireonice kol,
Fizark, Flauto Dolce, Flcelloguy, Flegelpuss, Flib, Flipping Mackerel, Flobbertthechicken, Flockmeal, Flubbit, Flyers5757, Foilfreeskier, Fonespiders, Forklift dense, Former user, Formeruser-8 1 ,
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Gabrielleitao, Gadfium, Gaff, Gaius Cornelius, Gaj2329, Galoubet, Games5522, GangofOne, GardenQuad, Garion96, Garrison Savannah, Gary King, GateKeeper, Gatos, Gatto Verde, Gchriss,
Gdm, GdoOl, Geeman, General Wesc, Generalstudent, Geni, George, George2001hi, GeorgeMoney, Georinia2, Ghe, Ghepeu, Ghjhy78, Giftlite, Gilisa, Gilliam, Gimmem ore time, GinaDana,
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Good Olfactory, Goplat, Gordonf238, Goudzovski, Gr8quizzer, GraemeL, Graham87, Grandpafootsoldier, Gravityguy, Grazon, Green caterpillar, GreenOeggs, GregAsche, Gregarchy,
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Halfmanl3, Hall Monitor, Hamsterlopithecus, Hanseichbaum, Happy-melon, HappyFrodo, Harachte, Harald88, Harbourcoat, Harlempiff, Harp, Harriseldon, Harry (usurped), Harryboyles,
Hashman28, Hawkestone, Hawkstein, Hazel77, Hbackman, Hdt83, Headbomb, HeartofaDog, Heartylunch, Heimstern, Hekerui, Hektor, HelloWorldToday, Hemlock Martinis, Hemmingsen,
Hengartn, Hephaestos, Here2fixCategorizations, HereToHelp, Herespenguin, HerkusMonte, Hermit crabs 101, Heron, Hibsch, Hillbrand, Hillman, Hires an editor, Historian932, Historymike,
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Integralolrivative, Inter, Intranetusa, InvisibleK, Inwind, Ioverka, Irdepesca572, Irishguy, Ironboyl 1, Isam, IsarSteve, Isis07, Island, IsmaelPR, Isnow, Isomorphic, IsultRose, It's-is-not-a-genitive,
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JamesReyes, Jameseavesjo, Jamesnakanda, JamibOy, Jamie63, Jan.Smolik, JanCeuleers, Janejellyroll, Jaoswald, Jappaji, Jaredroberts, Jaredwf, Jasperdoomen, Javal3690, Jaxl, Jay, Jay Gatsby,
Jayden54, Jayjg, Jc5202, Jcobb, JdH, Jdforrester, Jedidan747, Jeejee, Jeff Relf, Jefferson St. Charles, Jeffmcneill, Jeffsan, Jengod, Jennica, Jens Liebenau, Jermor, Jerry, JesseHogan, Jfpierce,
Jgeeky, Jhinman, Jhobsonl, Jibbajabba, Jibranl, Jiddisch, Jim Douglas, JimWae, Jimbo Donal Wales, Jimmyeatskids, Jitse Niesen, Jizef, Jkelly, Jkyrala, Jiang, Jlmlax99, Jnb, Jni, JoanneB,
Joaoantunes, JoeBlogsDord, JoeSmack, Joelr3 1 , Joeyadams, John, John Nevard, John Quincy Adding Machine, John Vandenberg, John of Lancaster, John254, JohnBlackburne, JohnDoe0007,
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Jwissick, K, Kl, KAMiKAZOW, KBi, KTC, Kaal, Kablammo, Kabuto Yakushi, Kadambarid, Kafkal251, Kaihsu, Kaisershatner, Kakofonous, Kaldari, Kane Franklin, Kane5187, Kappa,
Karafias, Karen Spain, Karl Dickman, KarlFrei, Karol Langner, Karppinen, Kasaalan, Kasparov, Katalaveno, Katimawan2005, Katzmik, Kaunascooliojr., Kawdyr, Kazuba, Kazvorpal,
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Khukri, Khushwantsingh987, Kigalil, King brosby, King of Hearts, KingRantheMan, Kingboyk, Kingforlday94, Kinneyboy90, KiwiJeff, Kiwiboy471, Kjetil r, KkmOlO, Klenod, Knarf-bz,
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Kurykh, Kv kreddy, Kwamikagami, Kwantus, Kwekubo, Kyle sb, L1A1 FAL, LJosil, LOKI 88, LTGioiorasaool, Labas321 123, Lacrimosus, LadyofShalott, Lamberson, Lampsalot, Lan360,
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Mandarax, Mann2, MarS, MarSch, Marc Mongenet, Marc-Andre ABbrock, MarcelB612, Marcika, Mariordo, MarkSutton, MarkSweep, Markcol , Markus Krotzsch, Markus Poessel, Marqus,
Martinp23, Martyman, Marwanl23, Mary Read, Marysunshine, Masako Kawasaki, Mashst24, Master of Puppets, MasterLycidas, Masterpiece2000, MathKnight, Mathemaxi, Matt00055,
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Kaufman, Michael Slone, Michael Zimmermann, Michael2, Michaelbusch, Michaelkulov, Miguel, Miguel Chong, MihaS, Mike Peel, Mike Rosoft, Mike holloweezy, Mike wiens, MikeVitale,
Mikesteam3 1 , MikeyTMNT, Milanov, Mild fuzz, Milk Coffee, Mindmatrix, Minesweeper, Mintguy, MisfitToys, Missmarple, Mistress Selina Kyle, Mitaphane, Mmernex, Mmoneypenny,
Modemac, Modernist, Modernrock2, Modster, Moe Epsilon, Molloy, Moncrief, Monegasque, Moni3, Monito, Monkeybuml, Montrealais, Moondyne, Moonlightchest, Moralis, Moreschi,
Moriori, Mormegil, Mortense, Mosaffa, MovGPO, Moxon, Mr Adequate, Mr Mulliner, Mr Tan, Mr. Granger, Mr. Yooper, Mr.adamg, Mr.grantevans2, MrDarcy, MrFish, MrMarmite, MrOllie,
Mrmdog, Mrmet31, Ms2ger, Mstroeck, Muchness, Mufka, Multifinderl7, Mushroom, Musical Linguist, MusoForde, Muzzle, Mvoltron, Mwanner, Mwelch, Mx3, Myrvin, Mysdaao, Mythole,
Mochtegern, N-true, N46, NBeale, NSR, NYScholar, Nacanacoo2, Nadavspi, Naddy, Nakon, Nameneko, Nanobug, Napoleon Dynamite42, Nastunya, Natalie Erin, Nathan8225, NathanHawking,
Natll, Naturalistic, Nauticashades, NawlinWiki, Nbauman, Ndenison, Neckro, Nectarflowed, Neddyseagoon, Nedvedfanl2, Neelix, Neilbeach, Neilc, Netoholic, Netpilot43556, Netscott,
Neurino, Neutrality, Nevl, NewEnglandYankee, Newport, Nicholas Tan, Nick, Nick Graves, Nick897, Nicop (Usurp), Night Gyr, Nihil novi, Nihiltres, Nikodemos, Nincbl23, Ninjagecko,
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Nova77, Npbierbrier, Nrbelex, Nuclear Warfare, Nuggetboy, Nurg, Nv8200p, OakMt, Oatmeal batman, Oberiko, ObsessiveMathsFreak, Ocolon, Ohad Asor, Ohconfucius, Ohnoitsjamie, Olathe,
Old Moonraker, Oldag07, OldakQuill, 01dlaptop321, Oldstylecharm, Oleg Alexandrov, Olessi, Olibroman, Olivemountain, Oliver Pereira, Olivier, Olleicua, Ollycity, Omfgthxkkbye,
OmicronperseiS, Onairdaprime, Onetwol, Onorem, Onthegogo, Oobopshark, Opelio, Optokinetics, Orionl 1M87, Ornil, Oroso, OsamaK, Oscarthecat, Ospalh, Otterpops, Ouishoebean,
OuroborosCobra, Overlord, OwenBlacker, OwenX, Oxnard27, P.M. Dahl, P4k, PFHLai, PIL1987, PJM, PKnight, PM Poon, Paine Ellsworth, PalestineRemembered, Pallabl234, Pallen,
Palma234@sympatico.ca, Pan Dan, Pandabear9876, Panser Born, Paranoid, Paranoid Eyes, Park70, Parrah, Pascal.Tesson, Patchouli, Pathoschild, Patstuart, Paul A, Paul August, Paul venter,
PaulTheOctopus, Paulleake, Paxsimius, Pb30, Pbech, Pde, Pdelong, Pearle, PedEyel, Pedro, PeepP, Penmachine, Peoplesunionpro, Pepsidrinka, Percy Snoodle, Perelaar, Perry Tachett, Persian
Poet Gal, Peruvianllama, Peter, Peter Ellis, Peter bertok, PeterCanthropus, PeterStJohn, Petersec, Peteryoungl44, Pethr, Petrus Steyn, Pevarnj, Pewwer42, Pfalstad, Pfortuny, Pgan002, Pgk,
Pgosta, Phelix06, Phenylalanine, Phil Boswell, PhilHibbs, Philip Trueman, Philthecow, Phoenix707, PhySusie, Physicists, Picaroon, Pictureuploader, Pietdesomere, Pigsonthewing, Pikiwedia,
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Poindexter Propellerhead, Pol098, Pol85, Politicaljunkie23, Pollinator, Pratyeka, Preplll, Presidentman, Presspiratehunter, Prichardson, Prithviraju 114, Prodego, ProfesOOl, Professor slats,
Progressivepantheist, Prolog, Przepla, PseudoOne, PseudoSudo, Pstevenson, Psy guy, PsyMar, Psycosonik2741, Punarbhava, Pyroclastic, Pyrospirit, Q0, QmunkE, Quadell, Quaon, Quasihuman,
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RJII, RNAi, RS1900, RaCha'ar, Rab V, Rabin, RachelBrown, Rad Racer, Radiantenergy, RadicalBender, Radio trucker 1983, Radius, Radzewicz, Raelus, Ragesoss, Rai31, Raj89er, Rajah, Ral315,
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Robomaeyhem, Robsteadman, Robth, Roby Wayne, Rodasmith, Roentgeniuml 1 1, Roger Anderton, Roger Hui, Rogerb67, Romanm, Rona2, Ronabop, RonaldFrits, Ronhjones, Rory096,
Rosarinagazo, RoyBoy, Royalguardl 1, Rrjanbiah, Rsabbatini, Rsage, Rtc, Ruakh, RunOrDie, Rursus, Russ, Russ kavalunah, RussBlau, Russianmissile, RxS, Ryandinhol4, Ryanrulz 11,
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Vimes, SamH, Samian, Samsara, Samuelsen, Sandstein, SandyGeorgia, Sangol23, Sankalpdravid, Santanielloa, Saotomi5102, Sarah, Saraneth, Saravan p, Sardanaphalus, Saruha, Satori Son,
Saturday, Savager, Savantl3, Save-Me-Oprah, Scarfacemz, Scarian, Scepia, Sceptre, Schfifty Three, Schizmatic, Schlafly, Schnufflus, Schutz, Schzmo, Scjessey, Scohoust, Scorchy, Scott Owen,
Scott3, Scottfisher, Scwlong, Scyker, Sean Whitton, Sean William, Sean.hoyland, Seansheep, Seattle Jorg, Seattle Skier, SebastianHelm, Secretlondon, Seglea, Selket, Separa, Septuagent,
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Singularity, Sir Richardson, SirFozzie, Sircus, Siryendor, Sjakkalle, SkerHawx, Skew-t, Skizzik, Skol fir, Sky Attacker, SkyWalker, Slark, Slempepperkake, Slicky, SlimboKimbo, Slreporter,
Smack, Snailwalker, Snowded, Snowolf, Snoyes, Snrcrackles, SomeGuyl 1 1 12, Someguyl221, Someone else, Soosed, Soumyasch, Soundcomm, SouthParkFan, Southwestspringroll, SpLoT,
SpNeo, Sparechange3, Spark, Spartaz, Spectre52, Spiksterl212, Spinnick597, Spirit Of Tiger, Spitfirel 1 11, Spittlespat, Splash, Sprewell, SpringZ, SpuriousQ, Spursnik, SquidSK, Squishy
bananas, Sreifa, Srich32977, Sriharsha2, Srijan89, Srikeit, Sriram 1991, Srleffler, StanleyOl 1, Star-of-David92, Starmagic, Starnold, Starx, Steeev, Steel, Steelmate, Steinbach, Stemonitis,
Stephan Schulz, Stephen G. Brown, Stephen e nelson, Stephenb, Stephencole2, Steve Quinn, Steve03Mills, Stevecolbert, Steveholt243, Steven Luo, Stevenj, Stevenmitchell, Stevertigo,
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SyntaxError55, Sysys, Syvanen, Szyslak, Slawomir Biaiy, T. Anthony, T1980, T@nn, TEB728, THEN WHO WAS PHONE?, TPickup, TUF-KAT, Ta bu shi da yu, TachyonOl, Taggart
Transcontinental, Tahiya, Tail, Tailpig, TakuyaMurata, Tamer han, Tanaats, Tangotango, Tarbecon, Targeman, Tarotcards, Tarquin, Tassedethe, Tastesoon, Tavilis, Taw, Tawker, Tbag88,
Tbjablin, Tcplanr, Ted87, TehWickerMan, Teleomatic, Tellyaddict, TenOfAllTrades, Teo2847, Tera557, Terence, Tevpg, TexasAndroid, Texture, The Anome, The Brain, The Encyclopedist,

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George Birkhoff Source: http://en.wikipedia.org/w/index.php?oldid=21860I99 Contributors: AHM, AO Charles, Algebraist, Anupamsr, Ardonik, Bender235, Berberisb, Bevildej, Btm, C S,
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Sheynhertz-Unbayg, Sholto Maud, Someone else, T-zero, TeaDrinker, The Tito, Todd Vierling, Tzartzam, Ypetrachenko, Zach, 37 anonymous edits

Stephen Smale Source: http://en.wikipedia.org/w/index.php?oldid=421 636796 Contributors: 0351456t, ALM scientist, Akriasas, Bender235, Bunnyhopll, C S, CJLL Wright, Charles
Matthews, Cmapm, CommonsDelinker, Curps, D6, David Eppstein, David Gerard, Dcoetzee, Debresser, Dgrant, Diamonddavej, Emerson7, Eruionnyron, Etacarl 1, Everyking, Freakofnurture,
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Yakov Sinai Source: http://en.wikipedia.org/w/index.php?oldid=35026804 Contributors: Charles Matthews, Ghirlandajo, Giftlite, Saga City, TheParanoidOne, 1 anonymous edits

Marston Morse Source: http://en.wiki pedia.org/ w/index.php?oldid=4 246806 3 9 Contributors: Amalas, Anestes, Bender235, Bunnyhopl 1 , C S, Charles Matthews, D6, Dan Gardner,
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G. A. Hedlund Source: http://en.wikipedia.org/w/index.php?oldid=231533155 Contributors: David Eppstein, Michael Hardy, SiamakT, Slarre, Waacstats

William Ross Ashby Source: http://en.wikipedia.org/w/index.php'?oldid=4221 19328 Contributors: AI, AndersFeder, Andre Engels, Bender235, Bradka, Crato, D6, Deodar, Erkan Yilmaz,
Fenice, Giftlite, Hpengwyn, Japanese Searobin, Jiuguang Wang, John b cassel, Jon Awbrey, Joyous!, Jwdietrich2, KYPark, Kate, Lexor, Liontooth, Loremaster, Macdonald-ross, Mais oui!,
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BorhanO, Canadian Paul, Chris Roy, Chuunen Baka, Cuthbertwong, D6, Darrell Greenwood, David Schaich, Docu, Download, Edward Lorenz, Ellywa, Erkan Yilmaz, Etacarl 1, Everyking,
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Hardy, Monegasque, Nbarth, OdedSchramm, Olessi, PDD, Retired username, Rjwilmsi, Rosiestep, 3 anonymous edits

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Benoit Mandelbrot Source: http://en.wikipedia.org/w/index.php?oldid=403961 106 Contributors: 0, 63.208.1 14.xxx, 75th Trombone, ASH1977LAW, Aboctok, Acornlord, Adam murgittroyd,
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